# Emergence of squeezed coherent states in Kaluza-Klein cosmology

A. S. Lemos<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
de Campina Grande, Caixa Postal 10071, 58429-900 Campina Grande, Paraíba,
Brazil  A. S. Pereira<EMAIL_ADDRESS>Instituto Federal do Pará,
68600-000 Bragança, Pará, Brazil  Departamento de Física, Universidade Federal
de Campina Grande, Caixa Postal 10071, 58429-900 Campina Grande, Paraíba,
Brazil  F. A. Brito<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande, Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil  Departamento de Física, Universidade Federal da
Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil  Joao R. L.
Santos<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
de Campina Grande, Caixa Postal 10071, 58429-900 Campina Grande, Paraíba,
Brazil

###### Abstract

In this work, we consider a propagating scalar field on Kaluza-Klein-type
cosmological background. It is shown that this geometrical description of the
Universe resembles - from a Hamiltonian standpoint - a damped harmonic
oscillator with mass and frequency, both time-dependents. In this scenario, we
construct the squeezed coherent states (SCSs) for the quantized scalar field
by employing the invariant operator method of Lewis-Riesenfeld (non-Hermitian)
in a non-unitary approach. The non-classicality of SCSs has been discussed by
examining the quadrature squeezing properties from the uncertainty principle.
Moreover, we compute the probability density, which allows us to investigate
whether SCSs can be used to seek traces of extra dimensions. We then analyze
the effects of the existence of supplementary space on cosmological particle
production in SCSs by considering different cosmological eras.

Kaluza-Klein cosmology, Squeezed coherent states, Non-unitary approach,
Integrals of motion

###### pacs:

04.50.+h, 03.65.Sq, 03.65.Fd, 03.65.Ca

##  I Introduction

The expansion of the Universe has been extensively studied in the literature
(Peebles2003, ; Oks2021, ), being mainly motivated by a substantial
observational and experimental basis (Riess1998, ; Perlmutter1999, ). Along
with cosmic radiation, Universe expansion is an essential component of the
$\Lambda$CDM cosmological model. Assuming that the Universe is homogeneous and
isotropic its geometrical description is given by the Friedmann-Lemaître-
Robertson-Walker (FLRW) cosmological background (Ellis1999, ). Furthermore,
from higher-dimensional embedding theories, cosmological models in a Kaluza-
Klein scenario have been presented as possible extensions to the FLRW model
(Leon1988, ; Wesson1994, ; Wesson2002, ; Bellini2003, ).

In its turn, the intersection of cosmology and quantum optics has been the
focus of increasing attention. As an example, we can highlight the problem of
cosmological particle creation in a time-dependent gravitational field
(Parker1968, ; Parker1969, ; Parker1971, ), which, afterwards, has been
addressed by the adoption of the language of squeezed states (SSs)
(Grishchuk1989, ; Grishchuk1990, ; Hu1994, ; Ford2021, ). In Refs.
(Grishchuk1989, ; Grishchuk1990, ), it was shown that the vacuum fluctuations
should produce relic gravitons, and, in its turn, in the course of
cosmological expansion would be squeezed into strongly SSs owing to the
interaction with the external gravitational field. This study has been carried
out considering the FLRW cosmological background and can provide valuable
insights into the physical conditions of the early Universe.

Although the classical and quantum propagation of scalar fields is well
developed in the context of the Minkowski metric (Parker2009, ; DeWitt1975, ),
its investigation into several background spacetime can provide important
information on the behavior of quantum systems in curved spacetime and their
possible mathematical framework similarity with another physical systems
(Pedrosa2021, ). Moreover, since the gravitational field universally couples
to all fields, it is interesting to investigate the quantum physical systems
in curved spacetime, even in the classical regime. Therefore, quantum optics
formalism has been applied to the study and construction of SSs of scalar
fields produced in several cosmological scenarios (Pedrosa2021, ; Bertoni1998,
; Pedrosa2004, ; Suresh2004, ; Suresh2004.2, ), in order to deal with quantum
effects in cosmology. In Ref. (Alencar2012, ), for instance, it was
investigated a quantized field propagating on $D$−dimensional de Sitter
spacetime and derived the corresponding Schrödinger states employing the
invariant operator method of Lewis-Riesenfeld (Lewis1969, ; Dodonov1975, ;
Dodonov2003, ).

In this work, we intend to investigate the effects of hidden dimensions on a
quantum system in a flat non-static spacetime background. So, we consider a
quantum scalar field propagating on a $5D$ Kaluza–Klein cosmological
background. Here, we assume that the gravitational field, produced by the
curvature of spacetime, is a classical background where a massless quantum
scalar field propagates. For our purposes, this problem may be reduced to a
damped harmonic oscillator with both mass and frequency time-dependent.
Furthermore, we employ the invariant operator method to construct time-
dependent SCSs by applying a non-unitary approach (Pereira2021, ; Pereira2023,
). These states satisfy the Schrödinger equation for the Hamiltonian of the
system associated with the scalar field. In its turn, in principle, the
inflaton field can be found in coherent states (CSs) and SSs beyond the
inflationary era (Suresh2004, ; Suresh2004.2, ) – at least when inflation
ends, and the scalar field begins to oscillate (Baumann2009, ). Thus, here we
intend to explore the possibility of finding the inflaton-like field in SCSs
during the radiation- and matter-dominated eras, aiming to seek for traces of
extra dimensions. For this purpose, we evaluate the transition probability
from a system initially prepared in a Fock state to the SCSs of a time-
dependent damped harmonic oscillator. The probability density has been
calculated and investigates the effects due to the extra dimension on this
quantity during different cosmic eras. Finally, we compute the time-dependent
number of particles produced in SCSs.We shown that the time-oscillation on the
number of created particles arises owing to the existence of an extra spatial
dimension.

The structure of this paper is divided in the following nutshell: section II
introduces a massless scalar field propagating in the Kaluza-Klein class of a
one-parameter $5D$ cosmological model. The quantum description of this system
and the discussion about the integral of motion method is the subject of
section III. In its turn, in section IV, we construct the SCSs and study the
non-classicality of the given system. Besides, we also have presented plots of
the probability density and discussed its results. In section V, we study the
particle production in the SCSs through the evolution in time of the number
operator. Section VI contains our final remarks and address insights for
future works.

##  II Propagating scalar field in Kaluza-Klein cosmologies

This section seeks to reformulate the quantum field propagating issue in a
$5D$ cosmological model such as a time-dependent oscillator. Let us start with
the Kaluza-Klein cosmological $5D$ line element given by (Wesson2002, )

$ds^{2}=\ell^{2}dt^{2}-A\left(t,\ell\right)\left(dx^{2}+dy^{2}+dz^{2}\right)-B\left(t\right)d\ell^{2},$
(1)

where $\ell$ is the extra-coordinate, and

$A\left(t,\ell\right)=t^{2/\alpha}\ell^{2/\left(1-\alpha\right)},\text{
and}\text{ \ }B\left(t\right)=\alpha^{2}\left(1-\alpha\right)^{-2}t^{2}.$ (2)

In the complementary $4D$ models, the nature of matter is defined by the
$\alpha$ parameter and encompasses both inflationary and non-inflationary
scenarios (Wesson2002, ). We point out that for $\alpha=3/2$, the Eq. (1)
describes the late Universe (matter era), while for $\alpha=2$ will describe
the early Universe (radiation era). Besides, one can consider
$\alpha\in\left(0,1\right)$ aiming to eventually describe inflationary
scenarios.

The Einstein-Hilbert action, which governs the theory of five-dimensional
gravity coupled to a scalar field in the $5D$ vacuum Universe can be written
as

$S=\int
dx^{5}\sqrt{\left|g\right|}\left[\frac{\hat{R}}{16\pi\hat{G}}+\mathcal{L}\left(\Phi,\Phi_{,M}\right)\right],$
(3)

where $g$ is the determinant of the metric tensor, $\hat{R}$ is the $5D$ Ricci
scalar, and $\hat{G}$ is the five-dimensional gravitational constant. In this
scenario, we restrict our attention to a minimally coupled real massless
scalar field $\Phi\left(\vec{X},t\right)$, with
$\vec{X}=\left(x,y,z,\ell\right)$, which has its dynamics governed by the
Lagrangian density

$\mathcal{L}=\frac{1}{2}g^{MN}\partial_{M}\Phi\partial_{N}\Phi,$ (4)

where $g^{MN}$ is the diagonal tensor metric given by the line element (1),
and capital Latin indices take values $0,1,2,3,4$. In this case we have a $5D$
Universe filled with a massless scalar field. Next, we may also decompose the
scalar field into a complete basis $u_{k}$, following the usual procedure
(e.g., (Pedrosa2021, ; Bertoni1998, ; Pedrosa2004, ; Alencar2012, )),

$\Phi\left(\vec{X},t\right)=\sum_{k}u_{k}\left(\vec{X}\right)\phi_{k}\left(t\right),$
(5)

where

$u_{k}\left(\vec{X}\right)=\frac{1}{\sqrt{\mathcal{V}}}e^{i\vec{k}\cdot\vec{X}},\qquad\phi_{k}\left(t\right)=\frac{1}{\sqrt{2}}\left(\phi_{k}^{1}+i\phi_{k}^{2}\right).$
(6)

Here, the amplitude $u_{k}$ is normalized in a finite volume $\mathcal{V}$,
and $\phi_{k}$ is written in terms of real and imaginary parts
$\left(i=1,2\right)$. This decomposition yields the action for the scalar
field

$S_{\phi}=\frac{1}{2}\sum_{k}\sum_{i=1,2}\int
dt\frac{\sqrt{A^{3}\left(t,\ell\right)B\left(t\right)}}{\ell}\left[\dot{\phi}_{k}^{i\,2}-\omega_{k}^{2}\left(t,\ell\right)\phi_{k}^{i\,2}\right],$
(7)

in the background (1), with

$\omega_{k}^{2}\left(t,\ell\right)\equiv\frac{k^{2}\ell^{2}}{A\left(t,\ell\right)}=\omega_{0k}^{2}\left(\tau/t\right)^{2/\alpha},\text{
\ }\omega_{0k}=k\,\ell^{\frac{\alpha}{\alpha-1}}\tau^{-1/\alpha},$ (8)

where $\tau$ is the initial time.

Henceforward, to avoid notational clutter, we suppress the $i$-index on the
scalar field. No confusion will arise because $k$ and $i$ are independent
indices (Pedrosa2021, ; Bertoni1998, ). It is now straightforward to define a
Hamiltonian for each $k$-mode of the scalar field from the action (7), such
that

$H_{k}=\frac{\Pi_{k}^{2}}{2m}+\frac{1}{2}m\omega_{k}^{2}\phi_{k}^{2},$ (9)

with

$\Pi_{k}=\frac{\partial
L}{\partial\dot{\phi}_{k}}=\sqrt{\frac{A^{3}\left(t,\ell\right)B\left(t\right)}{\ell}}\dot{\phi}_{k},\text{
\ }m=m_{0}\left(t/\tau\right)^{\frac{3+\alpha}{\alpha}},\text{ \
}m_{0}=\frac{\alpha\ell^{\frac{5+\alpha}{2\left(1-\alpha\right)}}}{\left|\alpha-1\right|}\tau^{\frac{3+\alpha}{\alpha}},$
(10)

where $\Pi_{k}$ is the conjugate momentum to the field $\phi_{k}$. The
classical equation of motion for $\phi_{k}$ can be obtained by varying the
action with respect to the field. Alternatively, turning our attention to Eq.
(9) we get the classical equation of motion

$\ddot{\phi}_{k}+\frac{3+\alpha}{\alpha
t}\dot{\phi}_{k}+\omega_{k}^{2}\phi_{k}=0,$ (11)

whose solution is

$\phi_{k}\left(t\right)=C_{1}t^{-3/2\alpha}J_{\nu}\left(\frac{kt^{\frac{\alpha-1}{\alpha}}\ell^{\frac{\alpha}{\alpha-1}}\alpha}{\left|\alpha-1\right|}\right)+C_{2}t^{-3/2\alpha}Y_{\nu}\left(\frac{kt^{\frac{\alpha-1}{\alpha}}\ell^{\frac{\alpha}{\alpha-1}}\alpha}{\left|\alpha-1\right|}\right),$
(12)

where $C_{1},C_{2}$ are real constants, $J_{\nu}$, and $Y_{\nu}$ are Bessel
functions of first and second kind, respectively, and

$\nu=\frac{3}{2\left|\alpha-1\right|}.$ (13)

Let us now identify the scalar field $\phi_{k}$ as an inflaton-like field. To
deal with this, it should be noted that, by assuming a coordinate
transformation such that $t=T^{\left(3-\alpha\right)/3}$, the Eq. (11) reduces
to

$\ddot{\phi}_{k}+\frac{3}{\alpha
T}\dot{\phi}_{k}+\widetilde{\omega}_{k}^{2}\phi_{k}=0,$ (14)

where
$\widetilde{\omega}_{k}=\frac{(\alpha-3)}{3}kT^{-\left(\alpha^{2}-\alpha+3\right)/3\alpha}\ell^{\alpha/\left(\alpha-1\right)}$,
and the overdot denotes partial derivative with respect to $T$. It is
important to highlight that from Eq. (14), we can identify the scalar field
$\phi_{k}$ as a $4D$ massive inflaton field with a quadratic potential

$V\left(\phi_{k}\right)=\frac{1}{2}\widetilde{\omega}_{k}^{2}\phi_{k}^{2}.$
(15)

Therefore, Eq. (14) characterizes the dynamics of the inflaton field during
inflation, radiation and matter eras. Furthermore, from Eqs. (11) and (14),
one can observe that the oscillation modes of scalar waves in the time-
dependent background can be analogously described as a parametric oscillator
system. In the following section, we employ this analogy and apply it to the
construction of SCSs for the quantized scalar field.

##  III Quantum description of the scalar field

In this section, we aimed to analyze the propagating quantum scalar field in a
Kaluza-Klein cosmological background from a quantum mechanics point-of-view.
In its turn, the quantum description of the system given by the Hamiltonian
(9) can be provided through canonical quantization, which in turn consists of
taking $\Pi_{k}\rightarrow\hat{\Pi}_{k}$, and
$\phi_{k}\rightarrow\hat{\phi}_{k}$, i.e., we promote the scalar field to an
operator $\hat{\phi_{k}}$ that satisfies the canonical commutation relations,

$\Pi_{k}\longrightarrow\hat{\Pi}_{k}=-i\hbar\frac{\partial}{\partial\phi_{k}},\text{
\ }\phi_{k}\longrightarrow\hat{\phi}_{k},\text{ \
}\left[\hat{\phi}_{k},\hat{\Pi}_{k^{\prime}}\right]=i\hbar\delta_{kk^{\prime}},$
(16)

where $\hat{\Pi}_{k}$ is the conjugate momentum to $\hat{\phi}_{k}$, and
$\delta_{kk^{\prime}}$ is the Kronecker delta. On its turn, we can represent
these operators in terms of the canonical creation and annihilation operators,
$\hat{a}_{k}^{\dagger}$ and $\hat{a}_{k}$, respectively, in the form

$\hat{\phi}_{k}=l_{k}\frac{\hat{a}_{k}+\hat{a}_{k}^{\dagger}}{\sqrt{2}},\text{
\
}\hat{\Pi}_{k}=\hbar\frac{\hat{a}_{k}-\hat{a}_{k}^{\dagger}}{i\sqrt{2}l_{k}},\text{
\ }\left[\hat{a}_{k},\hat{a}_{k}^{\dagger}\right]=1,$ (17)

where $l_{k}$ is a parameter with length dimension. Thus, in terms of the
transformations (16) and (17), we can quantize the Hamiltonian (9) into the
Hamiltonian $\hat{H}_{k}$, as follows

$\hat{H}_{k}=\frac{\hat{\Pi}_{k}^{2}}{2m}+\frac{1}{2}m\omega_{k}^{2}\hat{\phi}_{k}^{2}=\frac{\hbar\eta_{-}}{2}\left(\hat{a}_{k}^{2}+\hat{a}_{k}^{\dagger
2}\right)+\hbar\eta_{+}\hat{a}_{k}^{\dagger}\hat{a}_{k}+\frac{\hbar\eta_{+}}{2},$
(18)

where

$\displaystyle\hat{\Pi}_{k}^{2}=\frac{\hbar^{2}}{2l_{k}^{2}}\left(\hat{a}_{k}^{\dagger}\hat{a}_{k}+\hat{a}_{k}\hat{a}_{k}^{\dagger}-\hat{a}_{k}^{2}-\hat{a}_{k}^{\dagger
2}\right),\text{ \
}\hat{\phi}_{k}^{2}=\frac{l_{k}^{2}}{2}\left(\hat{a}_{k}^{\dagger}\hat{a}_{k}+\hat{a}_{k}\hat{a}_{k}^{\dagger}+\hat{a}_{k}^{2}+\hat{a}_{k}^{\dagger
2}\right),$
$\displaystyle\eta_{\pm}=\frac{l_{k}^{4}m^{2}\omega_{k}^{2}\pm\hbar^{2}}{2\hbar
ml_{k}^{2}}=\frac{l_{k}^{2}m_{0}\omega_{0k}^{2}}{2\hbar}\left(t/\tau\right)^{\frac{1+\alpha}{\alpha}}\pm\frac{\hbar}{2l_{k}^{2}m_{0}}\left(t/\tau\right)^{-\frac{3+\alpha}{\alpha}}.$
(19)

So, the time evolution of the quantum states $\left|\Psi_{k}\right\rangle$ is
given by the Schrödinger equation

$\displaystyle\hat{\Lambda}_{k}\left|\Psi_{k}\right\rangle=0,$ (20)
$\displaystyle\hat{\Lambda}_{k}=\hat{H}_{k}-i\hbar\partial_{t}.$ (21)

We can obtain the solution to (20) by applying the integral of motion method
(Lewis1969, ; Dodonov1975, ; Dodonov2003, ), which consists of defining a
time-dependent operator, for instance, $\hat{A}_{k}\left(t\right)$, which
commutes with $\hat{\Lambda}_{k}$, such that

$\frac{d\hat{A}_{k}}{dt}=\frac{i}{\hbar}\left[\hat{\Lambda}_{k},\hat{A}_{k}\right]=0.$
(22)

The eigenstates of $\hat{A}_{k}\left(t\right)$ must satisfy equation (20).
Following the proposal presented in (Pereira2021, ; Pereira2022, ;
Pereira2023, ), we will define $\hat{A}_{k}\left(t\right)$ in the form of a
linear combination of the operators $\hat{a}_{k}$ and $\hat{a}_{k}^{\dagger}$,
yielding to

$\displaystyle\hat{A}_{k}=f_{k}\hat{a}_{k}+g_{k}\hat{a}_{k}^{\dagger}+\varphi_{k},$
$\displaystyle\left[\hat{A}_{k},\hat{A}_{k}^{\dagger}\right]=\mu_{k},\text{ \
}\mu_{k}\equiv\left|f_{k}\right|^{2}-\left|g_{k}\right|^{2},$ (23)

where $f_{k}=f_{k}\left(t\right)$, $g_{k}=g_{k}\left(t\right)$, and
$\varphi_{k}=\varphi_{k}\left(t\right)$ are time-dependent complex functions.

Replacing the representations (18), (21), and (23) into (22), we get

$\displaystyle\dot{g}_{k}=i\eta_{-}f_{k}-i\eta_{+}g_{k},$
$\displaystyle\dot{f}_{k}=i\eta_{+}f_{k}-i\eta_{-}g_{k},$
$\displaystyle\dot{\varphi}_{k}=0.$ (24)

Given the initial conditions $f_{0k}=f_{k}\left(\tau\right)$,
$g_{0k}=g_{k}\left(\tau\right)$, and
$\varphi_{0k}=\varphi_{k}\left(\tau\right)$, we will have the following
solution for the system (24),

$\displaystyle f_{k}=$
$\displaystyle\frac{\pi\beta_{k}\tau^{\delta}}{4}\left[\frac{\tau^{\delta\left(\varepsilon-1\right)}}{t^{\delta\left(\varepsilon-1\right)}}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}F_{0k}+\frac{t^{\delta\varepsilon}}{\tau^{\delta\varepsilon}}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right]$
$\displaystyle+$ $\displaystyle\frac{i\pi\hbar\left(\tau
t\right)^{\delta\varepsilon+\frac{3+\alpha}{\alpha}}}{4l_{k}^{2}\delta
m}\left[W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}F_{0k}-\frac{l_{k}^{4}\delta^{2}\beta_{k}^{2}m^{2}}{\hbar^{2}\left(\tau
t\right)^{2\delta\varepsilon+\frac{7+\alpha}{\alpha}}}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right],$
$\displaystyle g_{k}=$
$\displaystyle\frac{\pi\beta_{k}\tau^{\delta}}{4}\left[\frac{\tau^{\delta\left(\varepsilon-1\right)}}{t^{\delta\left(\varepsilon-1\right)}}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}F_{0k}-\frac{t^{\delta\varepsilon}}{\tau^{\delta\varepsilon}}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right]$
$\displaystyle-$ $\displaystyle\frac{i\pi}{4}\frac{\hbar\left(\tau
t\right)^{\delta\varepsilon+\frac{3+\alpha}{\alpha}}}{l_{k}^{2}\delta
m}\left[W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}F_{0k}+\frac{l_{k}^{4}\delta^{2}\beta_{k}^{2}m^{2}}{\hbar^{2}\left(\tau
t\right)^{2\delta\varepsilon+\frac{7+\alpha}{\alpha}}}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right],$
(25)

where $F_{0k}$, $G_{0k}$ are constants determined by the initial conditions,
and
$W_{k,a,b}^{\left(\tau,t\right)}=J_{a}\left(\beta_{k}\tau^{\delta}\right)Y_{b}\left(\beta_{k}t^{\delta}\right)-J_{b}\left(\beta_{k}t^{\delta}\right)Y_{a}\left(\beta_{k}\tau^{\delta}\right)$.
For more details, please see Appendix A.

From here, it is convenient to define a new operator, in the form

$\hat{B}_{k}\equiv\frac{1}{f_{k}}\hat{A}_{k}=\hat{a}_{k}+\zeta_{k}\hat{a}_{k}^{\dagger}+\xi_{k},\text{
\
}\left[\hat{B}_{k},\hat{B}_{k}^{\dagger}\right]=1-\left|\zeta_{k}\right|^{2},$
(26)

where the time-dependent quantities $\xi_{k}$ and $\zeta_{k}$, are given by

$\xi_{k}\equiv\frac{\varphi_{k}}{f_{k}},\text{ \
}\zeta_{k}\equiv\frac{g_{k}}{f_{k}},$ (27)

which are identified as displacement and squeeze parameters, respectively, in
the non-unitary treatment of SCSs. In what follows, we can write $\hat{a}_{k}$
and $\hat{a}_{k}^{\dagger}$ in terms of the new operators $\hat{B}_{k}$ and
$\hat{B}_{k}^{\dagger}$, whose explicitly forms are

$\hat{a}_{k}=\frac{\hat{B}_{k}-\zeta_{k}\hat{B}_{k}^{\dagger}+\zeta_{k}\xi_{k}^{\ast}-\xi_{k}}{1-\left|\zeta_{k}\right|^{2}},\text{
\
}\hat{a}_{k}^{\dagger}=\frac{\hat{B}_{k}^{\dagger}-\zeta_{k}^{\ast}\hat{B}_{k}+\zeta_{k}^{\ast}\xi_{k}-\xi_{k}^{\ast}}{1-\left|\zeta_{k}\right|^{2}}.$
(28)

As we will see, from relations (28), we can obtain the average values and
uncertainty relation to the built SCSs.

##  IV Construction of SCSs in a non-unitary approach

The employ of quantum optics techniques, as the SSs formalism, has proven
highly advantageous for addressing numerous issues in cosmology
(Grishchuk1989, ; Grishchuk1990, ; Pedrosa2021, ; Bertoni1998, ; Pedrosa2004,
; Suresh2004, ). In its turn, the integrals of motion in a non-unitary
approach has shown itself as a suitable method for the study of time-dependent
quantum systems (Andrews1999, ; Pereira2021, ; Pereira2022, ; Pereira2023, ).
Therefore, by applying the non-unitary treatment, we intend to construct the
SCSs for this system. So, let us first define a non-unitary operator composed
by the displacement ($\xi_{k}$) and squeeze ($\zeta_{k}$) parameters, as we
see below

$\hat{S}_{k}=\exp\left(\xi_{k}\hat{a}_{k}^{\dagger}+\frac{1}{2}\zeta_{k}\hat{a}_{k}^{\dagger
2}\right).$ (29)

By applying the Baker–Campbell–Hausdorff theorem, i.e.,
$e^{A}Be^{-A}=B+\left[A,B\right]+2^{-1}\left[A,\left[A,B\right]\right]+...$,
it is possible to transform $\hat{B}_{k}$ into the operator $\hat{a}_{k}$, by
considering the relation

$\hat{a}_{k}=\hat{S}_{k}\hat{B}_{k}\hat{S}_{k}^{-1}.$ (30)

So, by applying Eq. (30) on the vacuum state $\left|0_{k}\right\rangle$, which
is annihilated by $\hat{a}_{k}$, i.e. $\hat{a}_{k}\left|0_{k}\right\rangle=0$,
we obtain

$\hat{B}_{k}\left|\xi_{k},\zeta_{k}\right\rangle=0,\text{ \
}\left|\xi_{k},\zeta_{k}\right\rangle=\Phi_{k}\exp\left(-\xi_{k}\hat{a}_{k}^{\dagger}-\frac{1}{2}\zeta_{k}\hat{a}_{k}^{\dagger
2}\right)\left|0_{k}\right\rangle,$ (31)

where $\Phi_{k}=\Phi_{k}\left(t\right)$ is a function to be determined such
that the states $\left|\xi_{k},\zeta_{k}\right\rangle$ simultaneously satisfy
the normalization condition and Eq. (20).

From the normalization condition applied to the states
$\left|\xi_{k},\zeta_{k}\right\rangle$, that is,
$\left\langle\zeta_{k},\xi_{k}|\xi_{k},\zeta_{k}\right\rangle=1$, we obtain,

$\left|\Phi_{k}\right|=\left(1-\left|\zeta_{k}\right|^{2}\right)^{1/4}\exp\left[-\frac{1}{2}\frac{\left|\xi_{k}\right|^{2}-\operatorname{Re}\left(\zeta_{k}\xi_{k}^{\ast
2}\right)}{1-\left|\zeta_{k}\right|^{2}}\right].$ (32)

In order to find the previous result, we used the relations below:

$\displaystyle\exp\left(2yz-z^{2}\right)={\displaystyle\sum\limits_{n=0}^{\infty}}\frac{H\left(y\right)}{n!}z^{n},\text{
\ }z=-\sqrt{\frac{\zeta_{k}}{2}}\hat{a}_{k}^{\dagger},\text{ \
}y=\frac{\xi_{k}}{\sqrt{2\zeta_{k}}},$
$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}H_{n}\left(x\right)H_{n}\left(y\right)\frac{z^{n}}{2^{n}n!}=\frac{1}{\sqrt{1-z^{2}}}\exp\left[\frac{2xyz-\left(x^{2}+y^{2}\right)z^{2}}{1-z^{2}}\right],$
$\displaystyle x=\frac{\xi_{k}^{\ast}}{\sqrt{2\zeta_{k}^{\ast}}},\text{ \
}y=\frac{\xi_{k}}{\sqrt{2\zeta_{k}}},\text{ \ }z=\left|\zeta_{k}\right|,$ (33)

where $H_{n}\left(x\right)$ are the Hermite polynomials. Thus, given that
$\Phi_{k}=\left|\Phi_{k}\right|e^{i\theta_{k}}$, the normalized states can be
written as

$\left|\xi_{k},\zeta_{k}\right\rangle=\left(1-\left|\zeta_{k}\right|^{2}\right)^{1/4}\exp\left(-\frac{1}{2}\frac{\left|\xi_{k}\right|^{2}-\zeta_{k}^{\ast}\xi_{k}^{2}}{1-\left|\zeta_{k}\right|^{2}}+i\theta_{k}\right)\exp\left(-\xi_{k}\hat{a}_{k}^{\dagger}-\frac{1}{2}\zeta_{k}\hat{a}_{k}^{\dagger
2}\right)\left|0_{k}\right\rangle.$ (34)

To determine $\theta_{k}$ we assumed that the states
$\left|\xi_{k},\zeta_{k}\right\rangle$ satisfy equation (20). For this
purpose, we must solve the equation below,

$\left\langle\zeta_{k},\xi_{k}\left|\hat{H}_{k}-i\hbar\partial_{t}\right|\xi_{k},\zeta_{k}\right\rangle=0.$
(35)

Since

$\left\langle\zeta_{k},\xi_{k}\left|\hat{H}_{k}\right|\xi_{k},\zeta_{k}\right\rangle=i\hbar\left\langle\zeta_{k},\xi_{k}\left|\partial_{t}\right|\xi_{k},\zeta_{k}\right\rangle+\hbar\dot{\theta}_{k}-\frac{\hbar\eta_{-}}{2}\operatorname{Re}\left(\zeta_{k}\right)+\frac{\hbar\eta_{+}}{2},$
(36)

where

$\displaystyle\left\langle\zeta_{k},\xi_{k}\left|\hat{H}_{k}\right|\xi_{k},\zeta_{k}\right\rangle=$
$\displaystyle\frac{\hbar\eta_{-}}{2}\frac{\left(\zeta_{k}\xi_{k}^{\ast}-\xi_{k}\right)^{2}+\left(\zeta_{k}^{\ast}\xi_{k}-\xi_{k}^{\ast}\right)^{2}-\left(\zeta_{k}+\zeta_{k}^{\ast}\right)\left(1-\left|\zeta_{k}\right|^{2}\right)}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}$
$\displaystyle+$
$\displaystyle\hbar\eta_{+}\frac{\left|\zeta_{k}\xi_{k}^{\ast}-\xi_{k}\right|^{2}+\left(1-\left|\zeta_{k}\right|^{2}\right)\left|\zeta_{k}\right|^{2}}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}+\frac{\hbar\eta_{+}}{2},$
(37)

and, by imposing that the states $\left|\xi_{k},\zeta_{k}\right\rangle$
satisfy the Schrödinger equation, we must have,

$\dot{\theta}_{k}=\frac{1}{2}\operatorname{Re}\left(\eta_{-}\zeta_{k}-\eta_{+}\right)\Longrightarrow\theta_{k}=\frac{1}{2}\int_{\tau}^{t}\operatorname{Re}\left(\eta_{-}\zeta_{k}-\eta_{+}\right)dt^{\prime}.$
(38)

Finally, the normalized states satisfying equations (20) and (31) take the
form

$\left|\xi_{k},\zeta_{k}\right\rangle=\left(1-\left|\zeta_{k}\right|^{2}\right)^{1/4}\exp\left(\frac{1}{2}\frac{\zeta_{k}^{\ast}\xi_{k}^{2}-\left|\xi_{k}\right|^{2}}{1-\left|\zeta_{k}\right|^{2}}+i\theta_{k}\right){\displaystyle\sum\limits_{n_{k}=0}^{\infty}}\frac{\left(-1\right)^{n_{k}}}{\sqrt{n_{k}!}}\left(\frac{\zeta_{k}}{2}\right)^{\frac{n_{k}}{2}}H_{n_{k}}\left(\frac{\xi_{k}}{\sqrt{2\zeta_{k}}}\right)\left|n_{k}\right\rangle.$
(39)

Thus, we have built the SCSs for the quantum field immersed in the
cosmological Kaluza-Klein background, whose dynamics is described by
Hamiltonian (18). The physical properties of these states will be investigated
in the following subsections.

Besides finding the normalized states, we can also determine the transition
probability $P_{n_{k}}=\left|\left\langle
n_{k}|\zeta_{k},\xi_{k}\right\rangle\right|^{2}$. In this case, we assume that
the system was initially prepared in a time-independent Fock state and will
suffer a transition to SCSs. Thus, we now can use the results (39) in order to
obtain the transition probability as follows:

$P_{n_{k}}=\sqrt{1-\left|\zeta_{k}\right|^{2}}\exp\left[\frac{\operatorname{Re}\left(\zeta_{k}^{\ast}\xi_{k}^{2}\right)-\left|\xi_{k}\right|^{2}}{1-\left|\zeta_{k}\right|^{2}}\right]\frac{\left|\zeta_{k}\right|^{n_{k}}}{2^{n_{k}}n_{k}!}\left|H_{n_{k}}\left(\frac{\xi_{k}}{\sqrt{2\zeta_{k}}}\right)\right|^{2}.$
(40)

Our next subsection will be dedicated to discuss the quadrature squeezing
properties of our system.

###  IV.1 Quadrature squeezing properties: Mean values and minimization of
uncertainty relation

In this subsection, in order to investigate the non-classicality of
constructed SCSs, we will analyze the quadrature squeezing properties from
uncertainty relations. Therefore, let us start by considering $\hat{\phi}_{k}$
and $\hat{\Pi}_{k}$ given in terms of the integrals of motion:

$\displaystyle\hat{\phi}_{k}=\frac{l_{k}}{\sqrt{2}}\frac{\left(1-\zeta_{k}^{\ast}\right)\hat{B}_{k}+\left(1-\zeta_{k}\right)\hat{B}_{k}^{\dagger}-2\operatorname{Re}\left[\left(1-\zeta_{k}^{\ast}\right)\xi_{k}\right]}{1-\left|\zeta_{k}\right|^{2}},$
$\displaystyle\hat{\Pi}_{k}=\frac{\hbar}{\sqrt{2}il_{k}}\frac{\left(1+\zeta_{k}^{\ast}\right)\hat{B}_{k}-\left(1+\zeta_{k}\right)\hat{B}_{k}^{\dagger}-2i\operatorname{Im}\left[\left(1+\zeta_{k}^{\ast}\right)\xi_{k}\right]}{1-\left|\zeta_{k}\right|^{2}}.$
(41)

where $l_{k}$ is a length-dimensional parameter which is related to the
initial standard deviation (Pereira2021, ). Using the relations (31) and (41),
we can easily calculate the mean values of the operators $\hat{\phi}_{k}$ and
$\hat{\Pi}_{k}$, whose explicitly forms are

$\displaystyle\overline{\phi_{k}}$
$\displaystyle=\overline{\phi_{k}}\left(t\right)=\left\langle\xi_{k},\zeta_{k}\left|\hat{\phi}_{k}\right|\zeta_{k},\xi_{k}\right\rangle=-\frac{\sqrt{2}l_{k}\operatorname{Re}\left[\left(1-\zeta_{k}^{\ast}\right)\xi_{k}\right]}{1-\left|\zeta_{k}\right|^{2}},$
$\displaystyle\overline{\Pi_{k}}$
$\displaystyle=\overline{\Pi_{k}}\left(t\right)=\left\langle\xi_{k},\zeta_{k}\left|\hat{\Pi}_{k}\right|\zeta_{k},\xi_{k}\right\rangle=-\frac{\hbar}{l_{k}}\frac{\sqrt{2}\operatorname{Im}\left[\left(1+\zeta_{k}^{\ast}\right)\xi_{k}\right]}{1-\left|\zeta_{k}\right|^{2}}.$
(42)

From here, we can identify that there is a correspondence between the squeeze
$\zeta_{k}$ and displacement $\xi_{k}$ parameters with the mean values of
$\overline{\phi_{k}}$ and $\overline{\Pi_{k}}$, such that

$\xi_{k}=-\frac{1+\zeta_{k}}{\sqrt{2}l_{k}}\overline{\phi_{k}}-\frac{il_{k}}{\hbar}\frac{1-\zeta_{k}}{\sqrt{2}}\overline{\Pi_{k}}.$
(43)

Besides by taking the square of $\hat{\phi}_{k}$ and $\hat{\Pi}_{k}$, we have

$\displaystyle\hat{\phi}_{k}^{2}$
$\displaystyle=\frac{l_{k}^{2}}{2}\frac{\left(1-\zeta_{k}^{\ast}\right)^{2}\hat{B}_{k}^{2}+2\left|1-\zeta_{k}\right|^{2}\hat{B}_{k}^{\dagger}\hat{B}_{k}+\left(1-\zeta_{k}\right)^{2}\hat{B}_{k}^{\dagger
2}}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}$
$\displaystyle-\frac{2l_{k}^{2}\operatorname{Re}\left[\left(1-\zeta_{k}^{\ast}\right)\xi_{k}\right]\left[\left(1-\zeta_{k}^{\ast}\right)\hat{B}_{k}+\left(1-\zeta_{k}\right)\hat{B}_{k}^{\dagger}\right]}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}+\frac{l_{k}^{2}}{2}\frac{\left|1-\zeta_{k}\right|^{2}}{1-\left|\zeta_{k}\right|^{2}}+\frac{2l_{k}^{2}\operatorname{Re}^{2}\left[\left(1-\zeta_{k}^{\ast}\right)\xi_{k}\right]}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}},$
$\displaystyle\hat{\Pi}_{k}^{2}$
$\displaystyle=\frac{2i\hbar^{2}\operatorname{Im}\left[\left(1+\zeta_{k}^{\ast}\right)\xi_{k}\right]\left[\left(1+\zeta_{k}^{\ast}\right)\hat{B}_{k}-\left(1+\zeta_{k}\right)\hat{B}_{k}^{\dagger}\right]}{l_{k}^{2}\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}$
$\displaystyle-\frac{\hbar^{2}}{2l_{k}^{2}}\frac{\left(1+\zeta_{k}^{\ast}\right)^{2}\hat{B}_{k}^{2}-2\left|1+\zeta_{k}\right|^{2}\hat{B}_{k}^{\dagger}\hat{B}_{k}+\left(1+\zeta_{k}\right)^{2}\hat{B}_{k}^{\dagger
2}}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}+\frac{\hbar^{2}}{2l_{k}^{2}}\frac{\left|1+\zeta_{k}\right|^{2}}{1-\left|\zeta_{k}\right|^{2}}+\frac{2\hbar^{2}\operatorname{Im}^{2}\left[\left(1+\zeta_{k}^{\ast}\right)\xi_{k}\right]}{l_{k}^{2}\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}.$
(44)

So, the previous results allow us to compute the mean values of
$\overline{\phi_{k}^{2}}$ and $\overline{\Pi_{k}^{2}}$, as follows

$\displaystyle\overline{\phi_{k}^{2}}$
$\displaystyle=\overline{\phi_{k}^{2}}\left(t\right)=\left\langle\xi_{k},\zeta_{k}\left|\hat{\phi}_{k}^{2}\right|\zeta_{k},\xi_{k}\right\rangle=\frac{l_{k}^{2}}{2}\frac{\left|1-\zeta_{k}\right|^{2}}{1-\left|\zeta_{k}\right|^{2}}+\frac{2l_{k}^{2}\operatorname{Re}^{2}\left[\left(1-\zeta_{k}^{\ast}\right)\xi_{k}\right]}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}},$
$\displaystyle\overline{\Pi_{k}^{2}}$
$\displaystyle=\overline{\Pi_{k}^{2}}\left(t\right)=\left\langle\xi_{k},\zeta_{k}\left|\hat{\Pi}_{k}^{2}\right|\zeta_{k},\xi_{k}\right\rangle=\frac{\hbar^{2}}{2l_{k}^{2}}\frac{\left|1+\zeta_{k}\right|^{2}}{1-\left|\zeta_{k}\right|^{2}}+\frac{2\hbar^{2}\operatorname{Im}^{2}\left[\left(1+\zeta_{k}^{\ast}\right)\xi_{k}\right]}{l_{k}^{2}\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}.$
(45)

Moreover, we are also able to calculate the standard deviation, i.e.,

$\displaystyle\sigma_{\phi_{k}}=\sigma_{\phi_{k}}\left(t\right)=\sqrt{\overline{\phi_{k}^{2}}-\overline{\phi_{k}}^{2}}=\frac{l_{k}}{\sqrt{2}}\frac{\left|1-\zeta_{k}\right|}{\sqrt{1-\left|\zeta_{k}\right|^{2}}},$
$\displaystyle\sigma_{\Pi_{k}}=\sigma_{\Pi_{k}}\left(t\right)=\sqrt{\overline{\Pi_{k}^{2}}-\overline{\Pi_{k}}^{2}}=\frac{\hbar}{\sqrt{2}l_{k}}\frac{\left|1+\zeta_{k}\right|}{\sqrt{1-\left|\zeta_{k}\right|^{2}}}.$
(46)

From Eqs. (46), one can see that standard deviations associated to the field
and momentum present the squeezing property.

Finally, one can find the Heisenberg uncertainty relation to the SCSs.
Therefore, in the hold of previous results, we can explicitly calculate the
product $\sigma_{\phi_{k}}\sigma_{\Pi_{k}}$, which results in

$\sigma_{\phi_{k}}\sigma_{\Pi_{k}}=\frac{\hbar}{2}\frac{\left|1-\zeta_{k}\right|\left|1+\zeta_{k}\right|}{1-\left|\zeta_{k}\right|^{2}}=\frac{\hbar}{2}\sqrt{1+\frac{4}{\hbar^{2}}\sigma_{\phi_{k}\Pi_{k}}^{2}},$
(47)

where $\sigma_{\phi_{k}\Pi_{k}}$ is the covariance given by

$\sigma_{\phi_{k}\Pi_{k}}=\frac{\left\langle\zeta_{k},\xi_{k}\left|\hat{\Pi}_{k}\hat{\phi}_{k}+\hat{\phi}_{k}\hat{\Pi}_{k}\right|\xi_{k},\zeta_{k}\right\rangle}{2}-\overline{\phi_{k}}\left(t\right)\overline{\Pi_{k}}\left(t\right)=-\hbar\frac{\operatorname{Im}\left(\zeta_{k}\right)}{1-\left|\zeta_{k}\right|^{2}}.$
(48)

Now, let us consider the minimization of the uncertainty relation at $t=\tau$.
For this, admit that

$\zeta_{0k}=\zeta_{k}\left(\tau\right)=\tanh\left(r\right)\Rightarrow
f_{0k}=\cosh\left(r\right),\text{ \ }g_{0k}=\sinh\left(r\right),\text{ \
}r\in\left(-\infty,\infty\right),$ (49)

where the $r$-parameter is related to the magnitude of squeeze parameter in
the initial time. Besides, the conditions (49) ensures that
$\mu_{0k}=\left|f_{0k}\right|^{2}-\left|g_{0k}\right|^{2}=1$. Our next step
consists in determining the parameter $l_{k}$ in terms of $r$ and
$\sigma_{\phi_{0k}}=\sigma_{\phi_{k}}\left(\tau\right)$. So, by taking the
relation (46) with $t=\tau$, we directly observe that

$l_{k}=\sqrt{2}e^{r}\sigma_{\phi_{0k}}.$ (50)

Then, from (48), and (49), we have
$\sigma_{\phi_{k}\Pi_{k}}\left(\tau\right)=0$, which implies that the relation
(47) is minimized at $t=\tau$, i.e.

$\sigma_{\phi_{0k}}\sigma_{\Pi_{0k}}=\frac{\hbar}{2},\text{ \
}\sigma_{\Pi_{0k}}=\sigma_{\Pi_{k}}\left(\tau\right).$ (51)

The condition found in Eq. (50) is going to by considered in our future
approaches.

Here, we intend to rewrite the uncertainty relations (46), aiming to show the
quadrature squeezing properties. For this, use the following dimensionless
transformation

$\hat{Q}_{k}=\frac{1}{l_{k}}\hat{\phi}_{k}=\frac{\hat{\phi}_{k}}{\sqrt{2}e^{r}\sigma_{\phi_{0k}}},\text{
\
}\hat{P}_{k}=\frac{l_{k}}{\hbar}\hat{\Pi}_{k}=\frac{\sqrt{2}e^{r}\sigma_{\phi_{0k}}}{\hbar}\hat{\Pi}_{k},\text{
\ }\left[\hat{Q}_{k},\hat{P}\right]=i.$ (52)

From here, we can rewrite (46) in the form

$\displaystyle\sigma_{Q_{k}}=\frac{1}{l_{k}}\sigma_{\phi_{k}}=\frac{1}{\sqrt{2}}\frac{\left|1-\zeta_{k}\right|}{\sqrt{1-\left|\zeta_{k}\right|^{2}}},$
$\displaystyle\sigma_{P_{k}}=\frac{l_{k}}{\hbar}\sigma_{\Pi_{k}}=\frac{1}{\sqrt{2}}\frac{\left|1+\zeta_{k}\right|}{\sqrt{1-\left|\zeta_{k}\right|^{2}}}.$
(53)

So, the Heisenberg uncertainty relation becomes

$\sigma_{Q_{k}}\sigma_{P_{k}}=\frac{1}{2}\sqrt{1+\frac{4}{\hbar^{2}}\sigma_{\phi_{k}\Pi_{k}}^{2}},\text{
\
}\sigma_{\phi_{k}\Pi_{k}}^{2}=\frac{\hbar^{2}\operatorname{Im}^{2}\left(\zeta_{k}\right)}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}}.$
(54)

(a)  (b)  (c)  (d)

Figure 1: A panel of the uncertainty relation plots which consider the
radiation era $\left(\alpha=2\right)$. Figs. (a) and (b) we set $r=0.5$, while
$\ell=10^{-2}$ and $\ell=1$, respectively. Figs. (c) and (d) we have fixed
$r=-0.5$, with $\ell=10^{-2}$ and $\ell=1$, respectively.

We should highlight that the uncertainties on the field and momentum depend on
the extra coordinate, $\ell$, and the $\alpha$-parameter that rules the
cosmological eras under consideration. In this case, in Figs. 1 and 2, we
present an investigation of the space of parameters considering the previously
uncertainties here derived [see Eq. (53)]. Therefore, as one can see, the
$r$-parameter is responsible for magnifying the squeeze properties on both
quadratures. The Figures 1(a) and 1(c) unveil that negative values of $r$
squeeze $\sigma_{P_{k}}$ standard deviation, while positive values squeeze
$\sigma_{Q_{k}}$. It is possible to verify that the variances for the SCSs are
lower than the generalized uncertainty minimum, confirming the non-
classicality of constructed states. On the other hand, the extra dimension
$\ell$ endows oscillatory pattern to the uncertainties, which could, in
principle, represent a signature of the extra dimension, as can be seen in
Figs. 1 and 2 [(b) and (d)]. Finally, the $\alpha$-parameter, which identifies
the cosmological era, gives rise to a bouncing behavior for the uncertainty
relations. So, in the matter era $\left(\alpha=3/2\right)$, we observe a slow
oscillation in the course its time evolution, compared to the radiation-
dominated era $\left(\alpha=2\right)$. We conclude this section by stressing
that the inherent characteristic of the SCSs, known as the squeezing property,
has been demonstrated from this analysis.

(a)  (b)  (c)  (d)

Figure 2: Uncertainty relations for $\alpha=3/2$ (matter era). Figs. (a) and
(b) we assume $r=0.5$, while $\ell=10^{-2}$ and $\ell=1$, respectively. Figs.
(c) $\ell=10^{-2}$ and (d) $\ell=1$ we consider $r=-0.5$.

###  IV.2 Representation of the SCSs in terms of the field $\phi_{k}$

Here, we intend to present the SCSs in another physical representation. With
this aim, we must follow some steps. First, we find the vacuum state
$\Psi_{0_{k}}\left(\phi_{k}\right)\equiv\left\langle\phi_{k}|0_{k}\right\rangle$
through the condition of annihilation
$\hat{a}_{k}\Psi_{0_{k}}\left(\phi_{k}\right)=0$ so that we use the
normalization condition, such that,

$\displaystyle\left(\frac{\phi_{k}}{l_{k}^{2}}+\frac{\partial}{\partial\phi_{k}}\right)\Psi_{0_{k}}\left(\phi_{k}\right)=0,$
$\displaystyle\Psi_{0_{k}}\left(\phi_{k}\right)=\frac{1}{\sqrt{l_{k}\sqrt{\pi}}}\exp\left(-\frac{\phi_{k}^{2}}{2l_{k}^{2}}\right),\text{
\ }\int_{-\infty}^{\infty}\Psi_{0_{k}}^{2}\left(\phi_{k}\right)d\phi_{k}=1.$
(55)

Therefore, by applying $n$ times the creation operator $\hat{a}_{k}^{\dagger}$
to $\Psi_{0}\left(\phi_{k}\right)$, we find the following form for the
$\Psi_{n}\left(\phi_{k}\right)$ states

$\Psi_{n_{k}}\left(\phi_{k}\right)=\frac{\left(\hat{a}_{k}^{\dagger}\right)^{n_{k}}}{\sqrt{n_{k}!}}\Psi_{0_{k}}\left(\phi_{k}\right)=\frac{H_{n_{k}}\left(\frac{\phi_{k}}{l_{k}}\right)}{\sqrt{l_{k}\sqrt{\pi}2^{n_{k}}n_{k}!}}\exp\left(-\frac{\phi_{k}^{2}}{2l_{k}^{2}}\right).$
(56)

Finally, from (39) and (56), we have the representation of SCSs in terms of
fields $\phi_{k}$,
$\Psi_{\zeta_{k},\xi_{k}}\left(\phi_{k}\right)\equiv\left\langle\phi_{k}|\zeta_{k},\xi_{k}\right\rangle$,
which is explicitly written as

$\displaystyle\Psi_{\zeta_{k},\xi_{k}}\left(\phi_{k},t\right)=\frac{\left(1-\left|\zeta_{k}\right|^{2}\right)^{1/4}}{\sqrt{\sqrt{2\pi}e^{r}\sigma_{\phi_{0k}}\left(1-\zeta_{k}\right)}}\exp\left[-\frac{1+\zeta_{k}}{4e^{2r}\sigma_{\phi_{0k}}^{2}\left(1-\zeta_{k}\right)}\left(\phi_{k}+\frac{2e^{r}\sigma_{\phi_{0k}}\xi_{k}}{1+\zeta_{k}}\right)^{2}\right.$
$\displaystyle\left.+\frac{\left(1+\zeta_{k}^{\ast}\right)\xi_{k}^{2}}{2\left(1+\zeta_{k}\right)\left(1-\left|\zeta_{k}\right|^{2}\right)}-\frac{\left|\xi_{k}\right|^{2}}{2\left(1-\left|\zeta_{k}\right|^{2}\right)}+i\theta_{k}\right].$
(57)

On the other hand, in terms of the $\overline{\phi_{k}}$ and
$\overline{\Pi_{k}}$ means values, we have

$\Psi_{\overline{\phi_{k}}}^{\sigma_{\phi_{k}}}\left(\phi_{k},t\right)=\frac{e^{i\vartheta_{k}}}{\sqrt{\sqrt{2\pi}\sigma_{\phi_{k}}}}\exp\left[-\frac{\left(\phi_{k}-\overline{\phi_{k}}\right)^{2}}{4\sigma_{\phi_{k}}^{2}}\right],$
(58)

where

$\displaystyle
e^{i\vartheta_{k}}=\left(\frac{1-\zeta_{k}^{\ast}}{1-\zeta_{k}}\right)^{1/4}\exp\left[\frac{i\sigma_{\phi_{k}\Pi_{k}}}{\hbar\sigma_{\phi_{k}}^{2}}\frac{\left(\phi_{k}-\overline{\phi_{k}}\right)^{2}}{2}+\frac{i\overline{\Pi_{k}}}{2\hbar}\left(2\phi_{k}-\overline{\phi_{k}}\right)+i\theta_{k}\right],$
$\displaystyle\overline{\phi_{k}}=-\frac{2e^{r}\sigma_{\phi_{0k}}\operatorname{Re}\left[\left(1-\zeta_{k}^{\ast}\right)\xi_{k}\right]}{1-\left|\zeta_{k}\right|^{2}},\text{
\
}\sigma_{\phi_{k}}=e^{r}\sigma_{\phi_{0k}}\frac{\left|1-\zeta_{k}\right|}{\sqrt{1-\left|\zeta_{k}\right|^{2}}},$
$\displaystyle\xi_{k}=\frac{4\hbar L_{k}\left(\tau
t\right)^{\frac{\varsigma-\delta}{2}}\varphi_{k}}{\pi\hbar\beta_{k}L_{k}\left[\tau^{\varsigma}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}e^{r}+t^{\varsigma}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}e^{-r}\right]+i\pi\beta_{k}\left[\hbar^{2}\left(\tau
t\right)^{\varsigma}W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}e^{r}-L_{k}^{2}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}e^{-r}\right]},$
$\displaystyle\zeta_{k}=\frac{\hbar
L_{k}\left[\tau^{\varsigma}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}e^{r}-t^{\varsigma}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}e^{-r}\right]-i\left[\hbar^{2}\left(\tau
t\right)^{\varsigma}W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}e^{r}+L_{k}^{2}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}e^{-r}\right]}{\hbar
L_{k}\left[\tau^{\varsigma}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}e^{r}+t^{\varsigma}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}e^{-r}\right]+i\left[\hbar^{2}\left(\tau
t\right)^{\varsigma}W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}e^{r}-L_{k}^{2}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}e^{-r}\right]},$
$\displaystyle\varsigma\equiv\delta\left(2\varepsilon-1\right),\text{ \
}L_{k}=\frac{2\delta
m_{0}\beta_{k}e^{2r}}{\tau^{\frac{3+\alpha}{\alpha}}}\sigma_{\phi_{0k}}^{2}.$
(59)

The representation found in Eq. (58) will allow us to discuss the probability
density, about which we will investigate the influence of both the extra
dimension $\ell$ and the cosmological era from the parameter $\alpha$.

####  IV.2.1 Probability density

Here we aim to study the probability density, which is defined as
$\rho_{\overline{\phi_{k}}}^{\sigma_{\phi_{k}}}\left(\phi_{k},t\right)\equiv\left|\Psi_{\overline{\phi_{k}}}^{\sigma_{\phi_{k}}}\left(\phi_{k},t\right)\right|^{2}$.
Thus, from Eq. (58), we can readily obtain

$\rho_{\overline{\phi_{k}}}^{\sigma_{\phi_{k}}}\left(\phi_{k},t\right)=\frac{1}{\sqrt{2\pi}\sigma_{\phi_{k}}}\exp\left[-\frac{\left(\phi_{k}-\overline{\phi_{k}}\right)^{2}}{2\sigma_{\phi_{k}}^{2}}\right],$
(60)

which is called the Gaussian or normal distribution and has been shown in
Figure 3. To depict these Figures we have assumed
$\hbar=1,\varphi_{k}=1,\sigma_{\phi_{0k}}=1,k=1,\tau=1$, and $r=0.5$. As we
see, once again, the extra-dimensional parameter is responsible for the
emergence of oscillatory behavior on the density, whereas the cosmological
parameter $\alpha$ will be related to the amplitude and frequency of
oscillations. Besides, small values for the $\ell$-parameter result in a
constant probability density in the time, while large values change
significantly its behavior, showing, in this case, an SCSs peak for initial
times ($t\sim 2$). In this case, it seems this physical system can be employed
to seek traces of extra dimensions.

(a)  (b)  (c)  (d)

Figure 3: Probability density contour plots $\left(t,\phi_{k}\right)$ for the
inflaton-like field considering the radiation era ($\alpha=2$), and matter era
($\alpha=3/2$).

##  V Particle creation in SCSs

Quite generically, the coupling of a scalar field to a curved background, in
general, yields to cosmological particle production (Parker1968, ; Parker1969,
; Parker1971, ; Parker2009, ). On its turn, the emergence of squeezed quantum
states is an unavoidable consequence whenever particle creation takes place
(Grishchuk1990, ). Therefore, in this section we aimed to investigate the
particle creation of the inflaton-like field in a Kaluza-Klein Universe
compatible with the SSs formalism from quantum optics. In this case, the
number of particles at a later time, $t$, created from the vacuum at the
initial time, $\tau$, is represented by the following equation:

$\overline{n}_{k}=\left\langle\zeta_{k},\xi_{k}\left|\hat{n}_{k}\right|\xi_{k},\zeta_{k}\right\rangle=\frac{\left|\zeta_{k}\right|^{2}\left(1-\left|\zeta_{k}\right|^{2}\right)+\left|\zeta_{k}\xi_{k}^{\ast}-\xi_{k}\right|^{2}}{\left(1-\left|\zeta_{k}\right|^{2}\right)^{2}},$
(61)

where $\hat{n}_{k}=\hat{a}^{\dagger}\hat{a}$ is the number operator, and the
state $\left|\xi_{k},\zeta_{k}\right\rangle$ is given by (39). The Eq. (61)
represents the average number of particles in the $k$-mode at time $t$.

As we have shown in Fig. 4, the number of created particles $\overline{n}_{k}$
is finite and will grow to a constant maximum value in time if the extra-
dimensional parameter $\ell\ll 1$. In this case, the displacement- and
$r$-parameters will be responsible to control this limit and, therefore, are
related to the excitation of the system. On the other hand, if $\ell\sim 1$,
we note that the number of created particles oscillates with time. Thus, the
extra dimension leaves a clear fingerprint on this physical observable, and
therefore, in principle, this physical system can be eventually applied to
impose constraints on the extra-dimensional parameter. In its turn, if
$\ell\ll 1$, we find that the number of produced particles in SCSs for the
matter era is greater than the one found in the radiation era.

Furthermore, we must highlight that Eq. (61) describes the number of particles
produced in the coherent states (CSs) for $\zeta_{k}=0$. This condition can be
determined from Eq. (24). Note that, in this case, we should have $g_{k}=0$.
However, such an imposition would imply that $f_{k}=0$ or $\eta_{-}=0$. Then,
in order to avoid the trivial solution, we must consider

$\eta_{-}=0\rightarrow t_{0}=\left(\frac{\hbar\left|\alpha-1\right|}{\alpha
kl_{k}^{2}\ell^{(\alpha-5)/2(\alpha-1)}}\right)^{\alpha/\left(\alpha+2\right)}.$
(62)

As we see, inflaton particles in CSs only are allowed at the initial time,
i.e., the time for which the uncertainty relation is minimized. Once that, in
this case, the number of particles depends on the extra-dimensional radius and
on the cosmological parameter. The constraints over the inflaton field in CSs
also may be a promising way to search for signatures of extra dimensions. On
the other hand, for $\xi_{k}=0$ ($\varphi_{k}=0$), Eq. (61) describes the
number of particles produced in the SSs.

(a)  (b)  (c)

Figure 4: Number of created particles in SCSs are shown for the case of a (a)
radiation and (b) matter eras of the Universe.

##  VI Final Remarks

In this work, we study a propagating massless scalar field on the Kaluza-
Klein-type cosmological background. In this context, by investigating the
Hamiltonian associated with the framework described in addition to the
respective equations of motion, one can found that this physical system
resembles a damped oscillator with both time-dependent mass and frequency.
Besides, we show that the $5D$ massless field can be identified with a massive
$4D$ inflaton field.

Following the SSs formalism of quantum mechanics, we have constructed the SCSs
for the quantized field in a non-unitary approach considering the invariant
operator method of Lewis-Riesenfeld. The uncertainty relations have been
investigated, and from a broad study of the parameter space, their
minimization was analyzed. Moreover, we have shown that the obtained states
have the squeezing property. We found that the extra-dimensional parameter,
$\ell$, is accountable for providing an oscillatory behavior presented on the
uncertainty relations, while the cosmological eras, controlled by the
$\alpha$-parameter, regulates the frequency of these oscillations. In this
case, for large values of the extra-dimensional parameter, the oscillations on
uncertainty relations give rise. This seems in total agreement with the fact
that $5D$ massless inflaton wave function can access our $4D$ Universe as we
increase the size of the extra-dimension. The probability density and
transition probability have been calculated, and again the extra dimension
accounts for raising the oscillatory behavior on these quantities. Moreover,
since the cosmological expansion can generate particles, we investigated
particle creation in SCSs in Kaluza-Klein cosmological scenario. By
considering a small value for the extra dimension, i.e., $\ell\ll 1$, the
number of created particles in SCSs reaches a maximum value. Besides, if we
work with higher values for this parameter, i.e., $\ell\sim 1$, the particle
production bounces in time. Finally, it is worth mentioning that once the
extra-dimensional parameter prints an unmistakable signature to all quantities
studied, perhaps this is a promising alternative physical system to seek
traces of hidden dimensions.

The methodology here presented can be implemented to several other geometrical
backgrounds, such as braneworld scenarios (Clifton2012, ). Moreover, these
discussions applied to the analogue gravity models (Barcelo2005, ) can provide
crucial insights into the Universe’s structure. Shortly, we expect to report
on some of these topics.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant No.
165/2018), for partial financial support. FAB acknowledges support from CNPq
(Grant No. 309092/2022-1). ASL acknowledges support from CAPES (Grant No.
88887.800922/2023-00). JRLS acknowledges CNPq (Grant No. 309494/2021-4), and
PRONEX/CNPq/FAPESQ-PB (Grant Nos. 165/2018, and 0015/2019) for financial
support. ASP thanks the support of the Instituto Federal do Pará.

##  Appendix A Integrals of motion ($f_{k}$ and $g_{k}$)

Let us start considering the differential equations

$\dot{f}_{k}=i\left(\eta_{+}f_{k}-\eta_{-}g_{k}\right),\text{ \
}\dot{g}_{k}=i\left(\eta_{-}f_{k}-\eta_{+}g_{k}\right).$ (63)

Now, by defining the new quantities

$F_{k}=f_{k}+g_{k}\Rightarrow\dot{F}_{k}=\dot{f}_{k}+\dot{g}_{k},\text{ \
}G_{k}=f_{k}-g_{k}\Rightarrow\dot{G}_{k}=\dot{f}_{k}-\dot{g}_{k},$ (64)

one may rewrite the equations (63) in the form

$\dot{F}_{k}=i\left(\eta_{+}+\eta_{-}\right)G_{k},\text{ \
}\dot{G}_{k}=i\left(\eta_{+}-\eta_{-}\right)F_{k}.$ (65)

On the other hand, knowing that

$\displaystyle\eta_{+}+\eta_{-}$
$\displaystyle=\frac{l_{k}^{2}m\omega_{k}^{2}}{\hbar}\Rightarrow\dot{\eta}_{+}+\dot{\eta}_{-}=\frac{1+\alpha}{\alpha
t}\left(\eta_{+}+\eta_{-}\right),$ $\displaystyle\eta_{+}-\eta_{-}$
$\displaystyle=\frac{\hbar}{l_{k}^{2}m}\Rightarrow\dot{\eta}_{+}-\dot{\eta}_{-}=-\frac{3+\alpha}{\alpha
t}\left(\eta_{+}-\eta_{-}\right),$ (66)

we get,

$\ddot{G}_{k}=i\left(\dot{\eta}_{+}-\dot{\eta}_{-}\right)F_{k}+i\left(\eta_{+}-\eta_{-}\right)\dot{F}_{k}=-\frac{3+\alpha}{\alpha
t}\dot{G}_{k}-\left(\eta_{+}^{2}-\eta_{-}^{2}\right)G_{k}.$ (67)

Furthermore, one can check that $\eta_{+}^{2}-\eta_{-}^{2}=\omega_{k}^{2}$.
Thus, we can rewritten (67) as follows

$\ddot{G}_{k}+\frac{3+\alpha}{\alpha t}\dot{G}_{k}+\omega_{k}^{2}G_{k}=0.$
(68)

From the Eq. (65), we find $F_{k}$, in the form

$F_{k}=-\frac{i}{\eta_{+}-\eta_{-}}\dot{G}_{k}=-\frac{il_{k}^{2}m}{\hbar}\dot{G}_{k}.$
(69)

Considering equations (68) and (69) with initial conditions
$F_{0k}=F_{k}\left(\tau\right)$ and $G_{0k}=G_{k}\left(\tau\right)$, we will
obtain

$\displaystyle
G_{k}=\frac{\pi\beta_{k}t^{\delta\varepsilon}}{2\tau^{\delta\left(\varepsilon-1\right)}}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}+\frac{i\pi\hbar\left(\tau
t\right)^{\delta\varepsilon+\frac{3+\alpha}{\alpha}}}{2l_{k}^{2}\delta
m}W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}F_{0k},$ $\displaystyle
F_{k}=\frac{\pi\beta_{k}\tau^{\delta\varepsilon}}{2t^{\delta\left(\varepsilon-1\right)}}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}F_{0k}-\frac{i\pi
l_{k}^{2}\delta\beta_{k}^{2}m}{2\hbar\left(\tau
t\right)^{\delta\varepsilon+\frac{4}{\alpha}}}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}G_{0k},$
(70)

where $\tau$ is the initial time and

$\displaystyle\varepsilon=\frac{3}{2\left|\alpha-1\right|},\text{ \
}\beta_{k}=\frac{\alpha\omega_{0k}\tau^{1/\alpha}}{\left|\alpha-1\right|},\text{
\ }\delta=\frac{\alpha-1}{\alpha},\text{ \
}W_{k,a,b}^{\left(\tau,t\right)}=J_{a}\left(\beta_{k}\tau^{\delta}\right)Y_{b}\left(\beta_{k}t^{\delta}\right)-J_{b}\left(\beta_{k}t^{\delta}\right)Y_{a}\left(\beta_{k}\tau^{\delta}\right),$
$\displaystyle
W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,\tau\right)}=\frac{2}{\pi\beta_{k}\tau^{\delta}},\text{
\ }W_{k,b,a}^{\left(\tau,t\right)}=-W_{k,a,b}^{\left(t,\tau\right)},\text{ \
}W_{k,a,a}^{\left(\tau,\tau\right)}=0.$ (71)

Finally, the functions $f_{k}$ and $g_{k}$ are determined from the relations
below

$f_{k}=\frac{F_{k}+G_{k}}{2},\text{ \ }g_{k}=\frac{F_{k}-G_{k}}{2}.$ (72)

Thus, for the $f_{k}$\- and $g_{k}$-functions, we get

$\displaystyle f_{k}=$
$\displaystyle\frac{\pi\beta_{k}\tau^{\delta}}{4}\left[\frac{\tau^{\delta\left(\varepsilon-1\right)}}{t^{\delta\left(\varepsilon-1\right)}}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}F_{0k}+\frac{t^{\delta\varepsilon}}{\tau^{\delta\varepsilon}}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right]$
$\displaystyle+$ $\displaystyle\frac{i\pi\hbar\left(\tau
t\right)^{\delta\varepsilon+\frac{3+\alpha}{\alpha}}}{4l_{k}^{2}\delta
m}\left[W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}F_{0k}-\frac{l_{k}^{4}\delta^{2}\beta_{k}^{2}m^{2}}{\hbar^{2}\left(\tau
t\right)^{2\delta\varepsilon+\frac{7+\alpha}{\alpha}}}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right],$
$\displaystyle g_{k}=$
$\displaystyle\frac{\pi\beta_{k}\tau^{\delta}}{4}\left[\frac{\tau^{\delta\left(\varepsilon-1\right)}}{t^{\delta\left(\varepsilon-1\right)}}W_{k,\varepsilon,\varepsilon-1}^{\left(\tau,t\right)}F_{0k}-\frac{t^{\delta\varepsilon}}{\tau^{\delta\varepsilon}}W_{k,\varepsilon,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right]$
$\displaystyle-$ $\displaystyle\frac{i\pi}{4}\frac{\hbar\left(\tau
t\right)^{\delta\varepsilon+\frac{3+\alpha}{\alpha}}}{l_{k}^{2}\delta
m}\left[W_{k,\varepsilon,\varepsilon}^{\left(\tau,t\right)}F_{0k}+\frac{l_{k}^{4}\delta^{2}\beta_{k}^{2}m^{2}}{\hbar^{2}\left(\tau
t\right)^{2\delta\varepsilon+\frac{7+\alpha}{\alpha}}}W_{k,\varepsilon-1,\varepsilon-1}^{\left(t,\tau\right)}G_{0k}\right],$
(73)

where

$f_{0k}=f_{k}\left(\tau\right)=\frac{F_{0k}+G_{0k}}{2},\text{ \
}g_{0k}=g_{k}\left(\tau\right)=\frac{F_{0k}-G_{0k}}{2}.$ (74)

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