# Squeezed coherent states for a free particle with time-varying mass

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
A. S. Lemos<EMAIL_ADDRESS>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

###### Abstract

We obtain the squeezed coherent states (SCS) for a free particle with
exponentially time-varying mass. We write these states in terms of the squeeze
and displacement parameters on the time-independent Fock states. Thus, we find
a condition on the displacement parameter such that the SCS can be considered
semiclassical states. We show that it is possible to obtain the coherent
states (CS) for a free particle with minimal uncertainty as long as the mass
increases with the time. We analyze the transition probability of a system
initially prepared in the time-independent Fock states to the free particle
SCS.

Coherent squeezed states, Nonunitary approach, Integrals of motion

###### pacs:

03.65.Sq, 03.65.Fd, 03.65.Ca

##  I Introduction

It is well-known that the motion of a free particle occurs when the sum of all
external forces is zero, which leads to the simplest physical system that we
know in nature. In this case, it is advantageous to use this model to
introduce some basic concepts, e.g., the wave packets representing a
semiclassical motion and the uncertainty relations (LIT1986, ; AND2008, ), in
introductory quantum mechanics text books. Furthermore, this system can be
seen as a limiting case of the harmonic oscillator when the frequency goes to
zero, which in quantum mechanics produces the correlated coherent states
(SHA1996, ) with applications in optics (SAL2007, ).

The coherent states (CS) are known as quantum states closer to a semiclassical
description for a given system under consideration. These states were obtained
in $1926$ by Schrödinger for the harmonic oscillator system (SCH1926, ), which
can be found in three equivalent ways: $(i)$ as eigenstates of the
annihilation operator, $(ii)$ as states generated by acting the displacement
operator on a preferred state (vacuum state), and $(iii)$ as states that
minimize the Heisenberg uncertainty relation (GLA1963, ; KLA1963, ; SUD1963,
). In turn, squeezed states (SS) form a class of nonclassical states, which
minimize the uncertainty relation. At the same time, these states present the
possibility of reducing the standard deviation of a physical quantity to a
value less than that calculated in the vacuum state at the expense of
increasing the standard deviation in another physical quantity (BAC2004, ;
SCH2017, ; DEY2021, ; ZEL2018, ). Both classes of states, CS and SS,
correspond to a particular case of the squeezed coherent states (SCS), see,
e.g., (DOD2003, ) and the references therein.

Unlike the harmonic oscillator, the free particle has a continuous energy
spectrum, which makes it difficult to introduce the ideas of annihilation and
creation operators. In addition, determining a preferred state for the
application of the displacement operator brings difficulties regarding the
normalization of these states, see (GEL2012, ). To circumvent these problems,
some authors address this simple quantum system through the integrals of
motion method, also known as the invariant operator method (AND1999, ;
BAG2014, ). This method consists in determining an operator that commutes with
the Schrödinger operator, aiming to obtain the eigenstates of this integral of
motion that must satisfy the Schrödinger equation. In Ref. (GUE2011, ),
classes of CS and SS have been constructed in terms of Hermite polynomials
using the quantum Arnold transformation. Recently a detailed study of
generalized CS was considered in Ref. (MAA2016, ), whereas symplectic
tomography were addressed in Ref. (OLG2022, ).

Following the recent paper (PER2021, ) – in which the SCS are constructed in a
nonunitary approach – we will obtain the SCS for a free particle with a mass
that varies exponentially with the time. The advantages of the nonunitary
approach compared to the unitary one for the construction of the SCS, lies in
some points: I - identification of the parameters $\xi$ and $\zeta$ (squeeze
and displacement parameters, respectively) with the functions $f$, $g$ and
$\varphi$, whose temporal evolution is determined by the construction of the
integrals of motion; II - as
$\hat{S}=\exp\left(\xi\hat{a}^{\dagger}+\frac{1}{2}\zeta\hat{a}^{\dagger
2}\right)$ depends only on $\hat{a}^{\dagger}$, it becomes advantageous to
obtain the condition under which the SCS satisfy the Schrödinger equation.
Furthermore, it is direct to found the relationship between the SCS and the
time-independent Fock states, which is not trivial when considering only the
integrals of motion, as discussed in Refs. BAG2014 ; BAG2015 . Following this
approach, we will be able to analyze the role of these parameters in the
calculation of uncertainty relations, as well as find conditions for which the
SCS can be considered semiclassical states. In addition, it will be possible
to evaluate the transition probability from a system initially prepared in a
Fock state to the SCS.

This work is structured as follows: In Sec. II, we construct the SCS and
discuss two limiting cases, i.e., $\zeta=0$, which will represent CS, while
$\xi=0$ describes SS. In Sec. III, we consider the SCS in the coordinate
representation and get the mean values of the coordinate and momentum, besides
their respective uncertainty relations. In the following, in Sec. IV, we find
the required conditions to be able to consider the SCS as a class of
semiclassical states. In Sec. V, we discuss the probability density and the
transition probability for the SCS. The final remarks are presented in Sec.
VI.

##  II Constructing the SCS of a free particle

In this section, we will show the way to be followed for the construction of
the SCS of a free particle. We will start by discussing the free particle
Hamiltonian, and then we will construct the integral of motion, i.e.,
operators that commute with Schrödinger operator. Finally, applying the
technique of the recent publication (PER2021, ), we will construct the SCS,
which are states annihilated by the integral of motion.

###  II.1 Free particle Hamiltonian

Let us consider the Hamiltonian of a free particle defined on real axis,
$x\in\left(-\infty,\infty\right)$, with time-dependent mass
$m\left(t\right)=m_{0}e^{\gamma t}$, as seen below

$\hat{H}\left(t\right)=\frac{\hat{p}^{2}}{2m\left(t\right)},\text{ \
}\hat{p}=-i\hbar\partial_{x},\text{ \ }\partial_{x}=\frac{\partial}{\partial
x},\text{ \
}\left[\hat{x},\hat{p}\right]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar,$ (1)

where $\gamma$ is a real parameter with frequency dimension, also known as
damping parameter. The Hamiltonian (1) can be taken as a particular case where
the frequency of the parametric oscillator of Caldirola-Kanai goes to zero
(CAL1941, ; KAN1948, ).

In turn, the quantum states $\left|\Psi\left(t\right)\right\rangle$, which
describe the time evolution of the system, should satisfy the Schrödinger
equation

$\hat{\Lambda}\left(t\right)\left|\Psi\right\rangle=0,\text{ \
}\hat{\Lambda}\left(t\right)=\hat{H}\left(t\right)-i\hbar\partial_{t},$ (2)

where $\hat{\Lambda}\left(t\right)$ is the Schrödinger operator. To simplify
the notation, we will omit the time dependence of the analyzed quantities.

Equivalently, we can represent $\hat{H}$ in terms of the annihilation
$\hat{a}$ and creation $\hat{a}^{\dagger}$ bosonic operators, as follows

$\hat{H}=\frac{\hbar^{2}}{l^{2}m}\frac{\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}-\hat{a}^{2}-\hat{a}^{\dagger
2}}{4},$ (3)

where $l$ is a real parameter with length dimension, and

$\displaystyle\hat{a}=\frac{1}{\sqrt{2}}\left(\frac{\hat{x}}{l}+\frac{il}{\hbar}\hat{p}\right),\text{
\
}\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}\left(\frac{\hat{x}}{l}-\frac{il}{\hbar}\hat{p}\right),$
$\displaystyle\hat{x}=l\frac{\hat{a}+\hat{a}^{\dagger}}{\sqrt{2}},\text{ \
}\hat{p}=\frac{\hbar}{l}\frac{\hat{a}-\hat{a}^{\dagger}}{i\sqrt{2}},\text{ \
}\left[\hat{a},\hat{a}^{\dagger}\right]=1.$ (4)

In turn, these operators act on the Fock states $\left|n\right\rangle$ in the
form,

$\displaystyle\hat{a}\left|n\right\rangle=\sqrt{n}\left|n-1\right\rangle,\text{
\
}\hat{a}^{\dagger}\left|n\right\rangle=\sqrt{n+1}\left|n+1\right\rangle,\text{
\
}\hat{n}\left|n\right\rangle=\hat{a}^{\dagger}\hat{a}\left|n\right\rangle=n\left|n\right\rangle,$
$\displaystyle\hat{a}\left|0\right\rangle=0,\text{ \
}\left|n\right\rangle=\frac{\left(\hat{a}^{\dagger}\right)^{n}}{\sqrt{n!}}\left|0\right\rangle,\text{
\ }n=0,1,2,...,$
$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}\left|n\right\rangle\left\langle
n\right|=1,\text{ \ }\left\langle
n^{\prime}|n\right\rangle=\delta_{n^{\prime},n},\text{ \
}\delta_{n^{\prime},n}=\left\\{\begin{array}[c]{c}1,\text{ \ if
}n^{\prime}=n\\\ 0,\text{ \ if }n^{\prime}\neq n\end{array}\right..$ (7)

In the construction of the SCS of the free particle with time-dependent mass,
we will consider its expansion on the Fock states. In this representation, we
can study the transition probability between harmonic-oscillator-type and the
free particle systems when considering the transition probability of a system
prepared in Fock states to the quantum states of the free particle. This
transition can be seen as the limit at which the oscillator frequency goes to
zero.

###  II.2 Integrals of motion as a linear combination of the operators
$\hat{a}$ and $\hat{a}^{\dagger}$

The integrals of motion method, also known as invariant operators, consists of
obtaining operators whose total derivative in time is zero. This condition
allows the construction of eigenstates common to integrals of motion and
Schrödinger operator. Furthermore, we have that the eigenvalue of the
integrals of motion must be a complex constant in time. This technique was
introduced by Lewis and Riesenfeld to study the time-dependent harmonic
oscillator (LEW1969, ). For more details and applications of this technique,
see, e.g., (DOD1975, ; BAG2015, ; PER2022, ).

Here, we define integrals of motion in the form of a linear combination of the
operators $\hat{a}$ and $\hat{a}^{\dagger}$, as seen below

$\displaystyle\hat{A}=f\hat{a}+g\hat{a}^{\dagger}+\varphi,$
$\displaystyle\left[\hat{A},\hat{A}^{\dagger}\right]=\mu,\text{ \
}\left|f\right|^{2}-\left|g\right|^{2}=\left|f_{0}\right|^{2}-\left|g_{0}\right|^{2}=\mu,$
(8)

where $f=f\left(t\right)$, $g=g\left(t\right)$ and
$\varphi=\varphi\left(t\right)$ are complex time-dependent functions with the
following initial conditions: $f_{0}=f\left(0\right)$, $g_{0}=g\left(0\right)$
and $\varphi_{0}=\varphi\left(0\right)$. The $\mu$-parameter is a real
constant defined in the range $\mu>0$. This parameter will be used to rewrite
the SCS in terms of the squeeze and displacement parameters, which will be
defined later. In this form, there is a correspondence between the SS and CS
of the group $SU\left(1,1\right)$ constructed by Peremolov (PER1986, ).

For operator $\hat{A}$ to be an integral of motion, it must satisfy the
following condition,

$\dot{\hat{A}}\equiv\frac{d\hat{A}}{dt}=\frac{i}{\hbar}\left[\hat{H},\hat{A}\right]+\frac{\partial\hat{A}}{\partial
t}=\frac{i}{\hbar}\left[\hat{\Lambda},\hat{A}\right]=0,$ (9)

At this point, it is interesting to highlight two properties arising from the
Eq. (9), which are: $\left(i\right)$ The total derivative of $\hat{A}$ is
zero, so the eigenvalue associated with this operator is a complex constant in
time; $\left(ii\right)$ Since the integral of motion commutes with the
Schrödinger operator, both operators have a common set of eigenstates.
Therefore, we can write

$\displaystyle\hat{A}\left|\Psi_{z,\lambda}\right\rangle=z\left|\Psi_{z,\lambda}\right\rangle,$
(10)
$\displaystyle\hat{\Lambda}\left|\Psi_{z,\lambda}\right\rangle=\lambda\left|\Psi_{z,\lambda}\right\rangle,$
(11)

where $\lambda=\lambda\left(t\right)$ is an arbitrary function on time, and
one can see that $\lambda=\lambda\left(t\right)=0$ reproduces the Schrödinger
equation (2).

Replacing the Eqs. (3) and (8) into (9), we obtain

$\left(\frac{\hbar f}{2l^{2}m}+\frac{\hbar
g}{2l^{2}m}-i\dot{g}\right)\hat{a}^{\dagger}-\left(\frac{\hbar
f}{2l^{2}m}+\frac{\hbar g}{2l^{2}m}+i\dot{f}\right)\hat{a}-i\dot{\varphi}=0.$
(12)

From (12), we must take the coefficients of operators $\hat{a}^{\dagger}$ and
$\hat{a}$ to be null, which leads to restrictions on the functions $f$, $g$,
and $\varphi$ in the form

$\dot{g}=-\frac{i\hbar}{2l^{2}m}\left(f+g\right),\text{ \
}\dot{f}=\frac{i\hbar}{2l^{2}m}\left(f+g\right),\text{ \ }\dot{\varphi}=0.$
(13)

Note that $\varphi\left(t\right)=\varphi\left(0\right)=\varphi$. The first two
equations in (13) yield the relation $f=-g+g_{0}+f_{0}$ that substituting it
into first Eq. (13), we determine the functions $g$, and $f$, as follows

$\displaystyle
g=g_{0}-\frac{i\hbar\left(f_{0}+g_{0}\right)}{2l^{2}m_{0}}\frac{1-e^{-\gamma
t}}{\gamma},$ $\displaystyle
f=f_{0}+\frac{i\hbar\left(f_{0}+g_{0}\right)}{2l^{2}m_{0}}\frac{1-e^{-\gamma
t}}{\gamma}.$ (14)

We must highlight that the operator $\hat{A}$ is parameterized in terms of the
integration constants $f_{0}$ and $g_{0}$. Thus, assuming different values for
these constants will produce new operators
$\hat{A}\longrightarrow\hat{A}\left(f_{0},g_{0},t\right)$, i.e., new integrals
of motion.

At this point, we can introduce new parameters $\xi=\xi\left(t\right)$ and
$\zeta=\zeta\left(t\right)$, as seen below

$\displaystyle\xi$
$\displaystyle=\frac{\varphi}{f}=\frac{2l^{2}m_{0}\gamma\xi_{0}}{2l^{2}m_{0}\gamma+i\hbar\left(1+\zeta_{0}\right)\left(1-e^{-\gamma
t}\right)},\text{ \ }\xi_{0}=\xi\left(0\right)=\frac{\varphi}{f_{0}},$ (15)
$\displaystyle\zeta$
$\displaystyle=\frac{g}{f}=\frac{2l^{2}m_{0}\gamma\zeta_{0}-i\hbar\left(1+\zeta_{0}\right)\left(1-e^{-\gamma
t}\right)}{2l^{2}m_{0}\gamma+i\hbar\left(1+\zeta_{0}\right)\left(1-e^{-\gamma
t}\right)},\text{ \ }\zeta_{0}=\zeta\left(0\right)=\frac{g_{0}}{f_{0}}.$ (16)

Thus, we can rewrite $f$ and $\mu$, in the form

$f=f_{0}\exp\left[\frac{i\hbar}{2l^{2}}\int_{0}^{t}\frac{1+\zeta\left(\tau\right)}{m\left(\tau\right)}d\tau\right],\text{
\
}\mu=\left|f\right|^{2}\left(1-\left|\zeta\right|^{2}\right)=\left|f_{0}\right|^{2}\left(1-\left|\zeta_{0}\right|^{2}\right).$
(17)

For the SCS in the nonunitary approach, the $\xi$,$\zeta$-parameters can be
interpreted as the displacement and squeeze parameters, respectively, see
(PER2021, ).

###  II.3 Time-dependent SCS of a free particle

In this section, we will construct the SCS in a nonunitary approach. In this
description, one can define the operator
$\hat{S}=\exp\left(\xi\hat{a}^{\dagger}+\frac{1}{2}\zeta\hat{a}^{\dagger
2}\right)$ and apply a transformation to the integrals of motion $\hat{A}$
that relates it directly to the bosonic annihilation operator $\hat{a}$, in
the form

$\hat{a}=\frac{1}{f}\hat{S}\hat{A}\hat{S}^{-1}.$ (18)

We find the transformation (18) through the Baker–Campbell–Hausdorff relation
$e^{A}Be^{-A}=B+\left[A,B\right]+\frac{1}{2}\left[A,\left[A,B\right]\right]+\ldots$.
As we will see, the application of nonunitary operator $\hat{S}$ simplifies
the calculation of $\Phi$-function, which ensures that the SCS satisfies the
Schrödinger equation. On the other hand, in the unitary approach, the operator
$\hat{S}$ introduces the operators $\hat{a}$ and $\hat{a}^{\dagger}$, which
requires additional effort in determining the $\Phi$-function due to the
commutation relations.

Considering the annihilation condition of the vacuum state
$\hat{a}\left|0\right\rangle=0$, we can write the relation (18) in the form

$\hat{A}\left|\xi,\zeta\right\rangle=0,$ (19)

where the general form of $\left|\xi,\zeta\right\rangle$ is given by

$\left|\xi,\zeta\right\rangle=\Phi\exp\left(-\xi\hat{a}^{\dagger}-\frac{1}{2}\zeta\hat{a}^{\dagger
2}\right)\left|0\right\rangle.$ (20)

Here, $\Phi=\Phi\left(t\right)$ is a complex function, which is determined by
imposing that the states $\left|\xi,\zeta\right\rangle$ satisfy the Eq. (2),
given by

$\Phi=C\exp\left(\frac{1}{2}\frac{\zeta^{\ast}\xi^{2}}{1-\left|\zeta\right|^{2}}-\frac{i\hbar}{4l^{2}}\int_{0}^{t}\frac{\zeta+1}{m}d\tau\right),$
(21)

where $C$ is a real constant. See Appendix A.

The imposition of the normalization condition
$\left\langle\zeta,\xi|\xi,\zeta\right\rangle=1$ allows us to determine the
constant $C$. See Appendix B. Therefore, the normalized states
$\left|\xi,\zeta\right\rangle$, which satisfy Eq. (2), have the following form

$\displaystyle\left|\xi,\zeta\right\rangle$
$\displaystyle=\left(1-\left|\zeta_{0}\right|^{2}\right)^{1/4}\exp\left(\frac{1}{2}\frac{\zeta^{\ast}\xi^{2}-\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}-\frac{i\hbar}{4l^{2}}\int_{0}^{t}\frac{\zeta+1}{m}d\tau\right)\exp\left(-\xi\hat{a}^{\dagger}-\frac{1}{2}\zeta\hat{a}^{\dagger
2}\right)\left|0\right\rangle$
$\displaystyle=\left(1-\left|\zeta_{0}\right|^{2}\right)^{1/4}\exp\left(\frac{1}{2}\frac{\zeta^{\ast}\xi^{2}-\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}-\frac{i\hbar}{4l^{2}}\int_{0}^{t}\frac{\zeta+1}{m}d\tau\right){\displaystyle\sum\limits_{n=0}^{\infty}}\frac{\left(-1\right)^{n}}{\sqrt{n!}}\left(\frac{\zeta}{2}\right)^{\frac{n}{2}}H_{n}\left(\frac{\xi}{\sqrt{2\zeta}}\right)\left|n\right\rangle.$
(22)

In what follows, we will refer the time-dependent states (22) as to SCS.

Equivalently, the states $\left|\xi,\zeta\right\rangle$ can be obtained by
directly considering the Eq. (19). For this, consider the decomposition of the
SCS in terms of the Fock states, in the form

$\left|\xi,\zeta\right\rangle={\displaystyle\sum\limits_{n=0}^{\infty}}c_{n}\left|n\right\rangle,$
(23)

where $c_{n}=c_{n}\left(t\right)$ are complex time-dependent functions.
Substituting the above expansion into Eq. (19), we obtain that

$c_{n+1}=-\frac{\sqrt{n}}{\sqrt{n+1}}\zeta
c_{n-1}-\frac{\xi}{\sqrt{n+1}}c_{n}.$ (24)

From the eq. (24) we can write $c_{n}$ in terms of $c_{0}$ employing a
recurrence relation.

We must highlight that obtaining a closed form for these coefficients is not
an easy task. In this sense, the nonunitary approach has shown itself a
practical mechanism to circumvent this problem. Another way to obtain the
closed form of $c_{n}$ is to identify Eq. (24) with the difference equation of
the Hermite polynomials, see (ZEL2021, ).

Taking into account the form of $c_{n}$, Eq. (23), becomes

$\left|\xi,\zeta\right\rangle=c_{0}{\displaystyle\sum\limits_{n=0}^{\infty}}\frac{\left(-1\right)^{n}}{\sqrt{n!}}\left(\frac{\zeta}{2}\right)^{\frac{n}{2}}H_{n}\left(\frac{\xi}{\sqrt{2\zeta}}\right)\left|n\right\rangle.$
(25)

It is interesting to note that $c_{0}$ is equivalent to the function $\Phi$
(see Eq. (20)), which was introduced as an ad hoc function, while $c_{0}$
arises naturally through the recurrence relation.

Applying the normalization condition
$\left\langle\xi,\zeta|\xi,\zeta\right\rangle=1$, and using the Mehler formula
for Hermite polynomials ERD1953 , we find the following form for $c_{0}$,

$c_{0}=\left(1-\left|\zeta\right|^{2}\right)^{1/4}\exp\left(-\frac{1}{2}\frac{\left|\xi\right|^{2}-\zeta^{\ast}\xi^{2}}{1-\left|\zeta\right|^{2}}+i\phi\right),$
(26)

where $\phi=\phi\left(t\right)$ must be a real function in order to guarantee
the normalization condition of the states $\left|\xi,\zeta\right\rangle$.
Imposing that the states $\left|\xi,\zeta\right\rangle$ satisfy the
Schrödinger equation (2), this implies that $\phi$ must have the form

$\phi=-\frac{\hbar}{4l^{2}}\int_{0}^{t}\frac{\operatorname{Re}\left(\zeta\right)+1}{m}d\tau.$
(27)

Therefore,

$\left|\xi,\zeta\right\rangle=\left(1-\left|\zeta\right|^{2}\right)^{1/4}\exp\left(\frac{1}{2}\frac{\zeta^{\ast}\xi^{2}-\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}+i\phi\right){\displaystyle\sum\limits_{n=0}^{\infty}}\frac{\left(-1\right)^{n}}{\sqrt{n!}}\left(\frac{\zeta}{2}\right)^{\frac{n}{2}}H_{n}\left(\frac{\xi}{\sqrt{2\zeta}}\right)\left|n\right\rangle.$
(28)

This result is the same as given in (22).

The overlap of the states $\left|\xi,\zeta\right\rangle$ for different squeeze
and displacement parameters is given by

$\displaystyle\left\langle\zeta_{1},\xi_{1}|\xi_{2},\zeta_{2}\right\rangle=\frac{\left(1-\left|\zeta_{2}\right|^{2}\right)^{1/4}\left(1-\left|\zeta_{1}\right|^{2}\right)^{1/4}}{\sqrt{1-\zeta_{1}^{\ast}\zeta_{2}}}\exp\left(\frac{1}{2}\frac{2\xi_{1}^{\ast}\xi_{2}-\zeta_{2}\xi_{1}^{\ast
2}-\zeta_{1}^{\ast}\xi_{2}^{2}}{1-\zeta_{1}^{\ast}\zeta_{2}}\right)$
$\displaystyle\times\exp\left[\frac{1}{2}\frac{\zeta_{2}^{\ast}\xi_{2}^{2}-\left|\xi_{2}\right|^{2}}{1-\left|\zeta_{2}\right|^{2}}+\frac{1}{2}\frac{\zeta_{1}\xi_{1}^{\ast
2}-\left|\xi_{1}\right|^{2}}{1-\left|\zeta_{1}\right|^{2}}-\frac{i\hbar}{4l^{2}}\int_{0}^{t}\frac{\operatorname{Re}\left(\zeta_{2}-\zeta_{1}\right)}{m}d\tau\right].$
(29)

This result shows that the SCS of a free particle are non-orthogonal. However,
these states satisfy to the completeness relation, see Appendix C,

$\displaystyle\int\left|\varphi,\zeta\right\rangle\left\langle\zeta,\varphi\right|d^{2}\varphi=1,$
$\displaystyle d^{2}\varphi=\left(\pi\mu\right)^{-1}du_{1}du_{2},\text{ \
}u_{1}=\operatorname{Re}\left(\varphi\right),\text{ \
}u_{2}=\operatorname{Im}\left(\varphi\right).$ (30)

This means that the SCS form an overcomplete basis in Hilbert space
$\mathcal{H}=L^{2}\left(\mathbb{R}\right)$ GAZ2009 . Indeed, the SCS form a
“continuous” family of quantum states in the Hilbert space $\mathcal{H}$,
labeled by all points of the complex plane $\mathbb{C}$.

We can get the CS and SS through the SCS by specifying the squeeze and
displacement parameters, as shown below:

a) Taking into account $\zeta=0\Rightarrow g=0$ we find the CS,

$\left|\xi,0\right\rangle=\exp\left(\frac{i\hbar}{4l^{2}m\gamma}-\frac{\left|\xi\right|^{2}}{2}\right)\exp\left(-\xi\hat{a}^{\dagger}\right)\left|0\right\rangle.$
(31)

The Eqs. (13) admit non-trivial solutions as long as we have the limit

$\gamma t\rightarrow\infty,$ (32)

which corresponds to a large mass particle. Thus, the displacement parameter
satisfies the following differential equation,

$\dot{\xi}=-\frac{i\hbar}{2l^{2}m}\xi\Longrightarrow\xi=\xi_{0}\exp\left(\frac{i\hbar}{2l^{2}m_{0}}\frac{e^{-\gamma
t}-1}{\gamma}\right).$ (33)

From (29), we have that

$\left\langle
0,\xi_{1}|\xi_{2},0\right\rangle=\exp\left(\xi_{1}^{\ast}\xi_{2}-\frac{1}{2}\left|\xi_{2}\right|^{2}-\frac{1}{2}\left|\xi_{1}\right|^{2}\right).$
(34)

It is easy to see that the CS are normalized by taking $\xi_{1}=\xi_{2}$ in
the above relation.

b) For SS we must impose the condition $\xi=0\Rightarrow\varphi=0$. Thus, we
find the following states,

$\left|0,\zeta\right\rangle=\left(1-\left|\zeta\right|^{2}\right)^{1/4}\exp\left(i\phi-\frac{1}{2}\zeta\hat{a}^{\dagger
2}\right)\left|0\right\rangle=\left(1-\left|\zeta\right|^{2}\right)^{1/4}\exp\left(i\phi\right){\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\zeta\right)^{n}\frac{\sqrt{\left(2n\right)!}}{2^{n}n!}\left|2n\right\rangle.$
(35)

From (29), we have that

$\left\langle\zeta_{1},0|0,\zeta_{2}\right\rangle=\frac{\left(1-\left|\zeta_{2}\right|^{2}\right)^{1/4}\left(1-\left|\zeta_{1}\right|^{2}\right)^{1/4}}{\sqrt{1-\zeta_{1}^{\ast}\zeta_{2}}}\exp\left[-\frac{i\hbar}{4l^{2}}\int_{0}^{t}\frac{\operatorname{Re}\left(\zeta_{2}-\zeta_{1}\right)}{m}d\tau\right].$
(36)

One can easily verify that the SS are normalized by taking
$\zeta_{2}=\zeta_{1}$. Except for the time evolution, the states (35) coincide
with the CS of the $SU\left(1,1\right)$ group when the Bargmann index is
$k=1/4$ and $\zeta\longrightarrow-\zeta^{\prime}$ (PER1986, ).

##  III Coordinate representation of the SCS

In this section, the SCS will be considered in the coordinate representation,
which will be useful to obtain the required conditions in order to ensure that
these states can be identified as semiclassical states. In addition, the mean
values of the coordinate and momentum will be obtained, as well as their
respective uncertainty relations.

The coordinate representation of the states $\left|\xi,\zeta\right\rangle$
will be found following some steps. First, we obtain the normalized solution
of the vacuum state that we find through the annihilation condition
$\left(\hat{a}\left|0\right\rangle=0\right)$, and which, in the coordinate
representation, becomes

$\displaystyle\hat{a}\Psi_{0}\left(x\right)=\frac{l}{\sqrt{2}}\left(\frac{x}{l^{2}}+\frac{\partial}{\partial
x}\right)\Psi_{0}\left(x\right)=0,\text{ \
}\Psi_{0}\left(x\right)\equiv\left\langle x|0\right\rangle,$
$\displaystyle\Psi_{0}\left(x\right)=\frac{1}{\sqrt{l\sqrt{\pi}}}\exp\left(-\frac{x^{2}}{2l^{2}}\right),\text{
\ }\int_{-\infty}^{\infty}\left|\Psi_{0}\left(x\right)\right|^{2}dx=1.$ (37)

Applying $n$ times creation operator $\hat{a}^{\dagger}$ in the state
$\Psi_{0}\left(x\right)$, we can determine
$\Psi_{n}\left(x\right)\equiv\left\langle x|n\right\rangle$, as seen below

$\Psi_{n}\left(x\right)=\frac{\left(\hat{a}^{\dagger}\right)^{n}}{\sqrt{n!}}\Psi_{0}\left(x\right)=\frac{H_{n}\left(\frac{x}{l}\right)}{\sqrt{l\sqrt{\pi}2^{n}n!}}\exp\left(-\frac{x^{2}}{2l^{2}}\right).$
(38)

In what follows, the SCS $\Psi_{\xi,\zeta}\left(x,t\right)\equiv\left\langle
x|\xi,\zeta\right\rangle$ in the coordinate representation takes the form

$\Psi_{\xi,\zeta}\left(x,t\right)=\frac{\left(1-\left|\zeta\right|^{2}\right)^{1/4}}{\sqrt{\sqrt{\pi}l\left(1-\zeta\right)}}\exp\left[-\frac{1+\zeta}{2l^{2}\left(1-\zeta\right)}\left(x+\frac{\sqrt{2}l\xi}{1+\zeta}\right)^{2}+\frac{\left(1+\zeta^{\ast}\right)\xi^{2}}{2\left(1+\zeta\right)\left(1-\left|\zeta\right|^{2}\right)}-\frac{\left|\xi\right|^{2}}{2\left(1-\left|\zeta\right|^{2}\right)}+i\phi\right].$
(39)

Here, we can obtain the CS by applying the condition $\zeta=0\Rightarrow g=0$.
Thus,

$\Psi_{\xi,0}\left(x,t\right)=\frac{1}{\sqrt{\sqrt{\pi}l}}\exp\left[-\frac{\left(x+l\sqrt{2}\xi\right)^{2}}{2l^{2}}+\frac{\xi^{2}}{2}-\frac{\left|\xi\right|^{2}}{2}+\frac{i\hbar}{4l^{2}m\gamma}\right].$
(40)

It is important to highlight that these states satisfy the Schrödinger
equation as long as we have the condition $\gamma t\rightarrow\infty$. On the
other hand, the SS are found when we consider $\xi=0\Rightarrow\varphi=0$, so
that

$\Psi_{0,\zeta}\left(x,t\right)=\frac{\left(1-\left|\zeta\right|^{2}\right)^{1/4}}{\sqrt{\sqrt{\pi}l\left(1-\zeta\right)}}\exp\left(-\frac{1+\zeta}{1-\zeta}\frac{x^{2}}{2l^{2}}+i\phi\right).$
(41)

###  III.1 Mean values, standard deviations and uncertainty relations

Now, we will consider the mean values, standard deviation, and uncertainty
relations for the coordinate and momentum. Given the condition (19), we can
easily calculate these quantities by rewriting the position and momentum in
terms of the integrals of motion. It follows from (4) and (8) that

$\displaystyle\hat{a}$
$\displaystyle=\frac{\hat{B}-\zeta\hat{B}^{\dagger}+\zeta\xi^{\ast}-\xi}{1-\left|\zeta\right|^{2}},\text{
\
}\hat{a}^{\dagger}=\frac{\hat{B}^{\dagger}-\zeta^{\ast}\hat{B}+\zeta^{\ast}\xi-\xi^{\ast}}{1-\left|\zeta\right|^{2}},\text{
\ }\hat{B}\equiv\frac{1}{f}\hat{A},$ $\displaystyle\hat{x}$
$\displaystyle=\frac{l}{\sqrt{2}}\frac{\left(1-\zeta^{\ast}\right)\hat{B}+\left(1-\zeta\right)\hat{B}^{\dagger}+2\operatorname{Re}\left(\zeta^{\ast}\xi-\xi\right)}{1-\left|\zeta\right|^{2}},$
$\displaystyle\hat{p}$
$\displaystyle=\frac{\hbar}{\sqrt{2}il}\frac{\left(1+\zeta^{\ast}\right)\hat{B}-\left(1+\zeta\right)\hat{B}^{\dagger}-2i\operatorname{Im}\left(\zeta^{\ast}\xi+\xi\right)}{1-\left|\zeta\right|^{2}}.$
(42)

Applying the condition (19), we can easily calculate the mean values of
$\hat{x}$ and $\hat{p}$,

$\displaystyle\overline{x}=\overline{x}\left(t\right)=\left\langle\zeta,\xi\left|\hat{x}\right|\xi,\zeta\right\rangle=-\frac{\sqrt{2}l\operatorname{Re}\left[\left(1-\zeta^{\ast}\right)\xi\right]}{1-\left|\zeta\right|^{2}}=\overline{x}_{0}+\frac{\overline{p}_{0}}{m_{0}}\frac{1-e^{-\gamma
t}}{\gamma},$
$\displaystyle\overline{p}=\overline{p}\left(t\right)=\left\langle\zeta,\xi\left|\hat{p}\right|\xi,\zeta\right\rangle=-\frac{\sqrt{2}\hbar}{l}\frac{\operatorname{Im}\left[\left(1+\zeta^{\ast}\right)\xi\right]}{1-\left|\zeta\right|^{2}}=-\frac{\sqrt{2}\hbar\operatorname{Im}\left[\left(1+\zeta_{0}^{\ast}\right)\xi_{0}\right]}{l\left(1-\left|\zeta_{0}\right|^{2}\right)}=\overline{p}_{0},$
$\displaystyle\overline{x}_{0}=-\frac{\sqrt{2}l}{1-\left|\zeta_{0}\right|^{2}}\operatorname{Re}\left[\left(1-\zeta_{0}^{\ast}\right)\xi_{0}\right]=-\frac{\sqrt{2}l\left|\xi_{0}\right|}{1-\left|\zeta_{0}\right|^{2}}\left[\cos\left(\theta_{\xi}\right)-\left|\zeta_{0}\right|\cos\left(\theta_{\xi}-\theta_{\zeta}\right)\right],\text{
\ }$
$\displaystyle\overline{p}_{0}=-\frac{\sqrt{2}\hbar\left|\xi_{0}\right|}{l\left(1-\left|\zeta_{0}\right|^{2}\right)}\left[\sin\left(\theta_{\xi}\right)+\left|\zeta_{0}\right|\sin\left(\theta_{\xi}-\theta_{\zeta}\right)\right],$
(43)

where $\zeta_{0}=\left|\zeta_{0}\right|e^{i\theta_{\zeta}}$ and
$\xi_{0}=\left|\xi_{0}\right|e^{i\theta_{\xi}}$. In what follows, we can
express the squeeze and displacement parameters in terms of the mean values
(43) in the form

$\xi=-\frac{1+\zeta}{\sqrt{2}}\frac{\overline{x}}{l}-\frac{1-\zeta}{\sqrt{2}}\frac{il\overline{p}}{\hbar}.$
(44)

Using (43), one can easily verify that the mean values $\overline{x}$ and
$\overline{p}$ satisfy the Hamilton equations

$\dot{x}=\frac{\partial H}{\partial p},\text{ \ }\dot{p}=-\frac{\partial
H}{\partial x},\text{ \ }H=\frac{p^{2}}{2m},$ (45)

where $H$ is the classical Hamiltonian that corresponds to the quantum
Hamiltonian (1). Thus, the pair $\overline{x}$ and $\overline{p}$ represents a
classical trajectory in the phase-space of the system under consideration. All
such trajectories can be parameterized by the initial conditions
$\overline{x}_{0}$ and $\overline{p}_{0}$.

In its turn, the standard deviations of $\overline{x}$ and $\overline{p}$ are
shown below

$\displaystyle\sigma_{x}$
$\displaystyle=\sigma_{x}\left(t\right)=\sqrt{\overline{x^{2}}-\overline{x}^{2}}=\frac{l}{\sqrt{2}}\frac{\left|1-\zeta\right|}{\sqrt{1-\left|\zeta\right|^{2}}},$
$\displaystyle\sigma_{p}$
$\displaystyle=\sigma_{p}\left(t\right)=\sqrt{\overline{p^{2}}-\overline{p}^{2}}=\frac{\hbar}{\sqrt{2}l}\frac{\left|1+\zeta\right|}{\sqrt{1-\left|\zeta\right|^{2}}}.$
(46)

From here, we can obtain the Heisenberg uncertainty relation

$\sigma_{x}\sigma_{p}=\frac{\hbar}{2}\frac{\left|1-\zeta\right|\left|1+\zeta\right|}{1-\left|\zeta\right|^{2}}=\frac{\hbar}{2}\sqrt{1+\frac{4\operatorname{Im}^{2}\left(\zeta\right)}{\left(1-\left|\zeta\right|^{2}\right)^{2}}}.$
(47)

The uncertainty relation shows that for any time, the inequality
$\sigma_{x}\sigma_{p}\geq\frac{\hbar}{2}$ is always satisfied. Moreover, the
equality is found when the squeeze parameter is a real quantity, i.e.,
$\operatorname{Im}\left(\zeta\right)=0$. Note also that, in the limit
$\zeta\rightarrow\pm 1$ we can to obtain values of $\sigma_{x}$ or
$\sigma_{p}$ lower than $\sqrt{\hbar/2}$, which characterizes the squeezing
property of the SS.

Now, consider the covariance $\sigma_{xp}$,

$\sigma_{xp}=\frac{1}{2}\left\langle\zeta,\xi\left|\left(\hat{x}-\overline{x}\right)\left(\hat{p}-\overline{p}\right)+\left(\hat{p}-\overline{p}\right)\left(\hat{x}-\overline{x}\right)\right|\xi,\zeta\right\rangle=-\hbar\frac{\operatorname{Im}\left(\zeta\right)}{1-\left|\zeta\right|^{2}}.$
(48)

In what follows, we can calculate the Robertson-Schrödinger uncertainty
relation (ROB1930, ),

$\sigma_{x}^{2}\sigma_{p}^{2}-\sigma_{xp}^{2}=\frac{\hbar^{2}}{4}.$ (49)

This result shows that the SCS of a free particle corresponds to a class of
correlated CS (DOD1980, ).

In general, the uncertainty relation is not minimized when evaluated for the
free particle. However, at $t=0$, we can obtain a condition on the parameter
$l$ such that the uncertainty relation is minimized. Initially, let us rewrite
the constants $f_{0}$ and $g_{0}$ in the form

$f_{0}=\cosh\left(r\right),\text{ \
}g_{0}=\sinh\left(r\right)\Rightarrow\zeta_{0}=\tanh\left(r\right),\text{ \
}-\infty<r<\infty.$ (50)

These conditions imply that
$\mu=\left|f_{0}\right|^{2}-\left|g_{0}\right|^{2}=1$. Next, taking the
standard deviation at $t=0$, one can rewrite the parameter $l$, as seen below

$\sigma_{x_{0}}=\sigma_{x}\left(0\right)=\frac{l}{\sqrt{2}}\frac{1-\zeta_{0}}{\sqrt{1-\zeta_{0}^{2}}}\Rightarrow
l=\sqrt{2\frac{1+\zeta_{0}}{1-\zeta_{0}}}\sigma_{x_{0}}=\sqrt{2}e^{r}\sigma_{x_{0}},$
(51)

where $\sigma_{x_{0}}=\frac{\hbar}{2\sigma_{p_{0}}}$. We will take this
condition of minimization of the uncertainty relation in the next sections.

We shall now discuss the time evolution for the Heisenberg uncertainty
relation since we already know $\sigma_{x}$ and $\sigma_{p}$ explicitly.
Calculating the product of the uncertainties on the position and moment, we
get

$\sigma_{x}\sigma_{p}=\frac{\hbar}{2}\sqrt{1+\frac{\hbar^{2}}{4m_{0}^{2}\sigma_{x_{0}}^{4}}\left(\frac{1-e^{-\gamma
t}}{\gamma}\right)^{2}}.$ (52)

The Figure 1 shows the plot of Eq. (52). As we see, if $\gamma>0$, the
Heisenberg uncertainty relation increases with the time and presents a plateau
region (finite value) as $t$ goes to infinity,
$\sigma_{x}\sigma_{p}=\hbar/2\sqrt{1+\hbar^{2}/4m_{0}^{2}\gamma^{2}\sigma_{x_{0}}^{4}}$.
In the regime $\gamma t\ll 1$, we get
$\sigma_{x}\sigma_{p}=\hbar/2\sqrt{1+\hbar^{2}t^{2}/4m_{0}^{2}\sigma_{x_{0}}^{4}}$,
which reproduce the result obtained in Ref. (LIT1986, ). In particular, this
equation shows us the spreading property of the free particle, similarly to
what happens in wave packets. On the other hand, if $\gamma<0$, the
corresponding Heisenberg uncertainty relation goes asymptotically in time as
$\sim e^{\left|\gamma\right|t}/\gamma$
$\left(\left|\gamma\right|t\rightarrow\infty\right)$.

(a)

(b)

Figure 1: The time evolution of free particle Heisenberg uncertainty relation.
We have here fixed $\hbar=1$, $m_{0}=1$, $\sigma_{x_{0}}=1$. We have assumed
$\gamma>0$ in (a), and $\gamma<0$, in (b).

Aiming to discuss the squeeze properties of SCS, we consider the quadratures
$\hat{Q}$ and $\hat{P}$, which one can be obtained by applying the
dimensionless transformation,

$\displaystyle\hat{Q}\equiv\hat{x}l^{-1}=\frac{1}{\sqrt{2}e^{r}}\frac{\hat{x}}{\sigma_{x_{0}}},\text{
\
}\hat{P}\equiv\frac{l}{\hbar}\hat{p}=\sqrt{2}e^{r}\frac{\sigma_{x_{0}}\hat{p}}{\hbar},\text{
\ }\left[\hat{Q},\hat{P}\right]=i,$
$\displaystyle\tau\equiv\frac{\hbar}{m_{0}l^{2}}t=\frac{\hbar}{2m_{0}e^{2r}\sigma_{x_{0}}^{2}}t=e^{-2r}\gamma
t\Longrightarrow\gamma t=e^{2r}\tau,\text{ \
}\gamma\equiv\frac{\hbar}{2m_{0}\sigma_{x_{0}}^{2}},$
$\displaystyle\sigma_{Q}=\frac{1}{l}\sigma_{x}=\frac{1}{\sqrt{2}}e^{-r}\sqrt{1+\left(1-e^{-e^{2r}\tau}\right)^{2}},\text{
\ }\sigma_{P}=\frac{l}{\hbar}\sigma_{p}=\frac{1}{\sqrt{2}}e^{r}.$ (53)

(a)

(b)

Figure 2: Quadrature squeezing was obtained by considering $r=-0.4$ in 2a and
$r=0.4$ in 2b.

An estimate of the nonclassicality of this system is quadrature squeezing,
shown in graphs 2. In Fig. 2a, it is shown that the standard deviation
$\sigma_{P}$ is below the threshold
$\sigma_{Q_{0}}^{2}=\sigma_{P_{0}}^{2}=1/2$ for values $r<0$ and remains
unchanged with time evolution. On the other hand, for $r>0$, we get that the
standard deviation $\sigma_{Q}$ is below the reference value $1/2$ and goes to
a constant value for $\tau\gg 1$, see Fig. 2b. Since the particle has an
exponentially time-dependent mass, $\sigma_{Q}$ grows with time until it
reaches a constant value in all the cases. Besides, in both the cases ($r<0$
and $r>0$), only to $\tau=0$ we get the minimized uncertainty relation.

##  IV SCS as semiclassical states

Here, we will obtain conditions under which we can consider the SCS as a class
of semiclassical states. These states are characterized so that when averaging
physical quantities such as position and momentum, we find that they evolve
along the classical trajectory. In addition, the standard deviation for each
of these means must have a well-behaved range of variation.

In general, the SCS of a free particle cannot be considered semiclassical
states, since the standard deviation in the coordinate increases with time, in
the form

$\overline{x}=\overline{x}_{0}+\frac{\overline{p}_{0}}{m_{0}}\frac{1-e^{-\gamma
t}}{\gamma},\text{ \
}\sigma_{x}=\sigma_{x_{0}}\sqrt{1+\frac{\hbar^{2}}{4m_{0}^{2}\sigma_{x_{0}}^{4}}\left(\frac{1-e^{-\gamma
t}}{\gamma}\right)^{2}}.$ (54)

Semiclassical motion can be characterized by the change slowly with time $t$
in the form of the standard deviation (54). Notice that the form of
$\sigma_{x}$ changes owing to the change in the quantity
$\frac{\hbar^{2}}{4m_{0}^{2}\sigma_{x_{0}}^{4}}\left(\frac{1-e^{-\gamma
t}}{\gamma}\right)^{2}$ with time. If we assume that the quantity responsible
for the change in $\sigma_{x}$ is significantly smaller than the distance
traveled by the particle in the same amount of time, the form of the standard
deviation (54) changes slowly, which characterize, in a certain sense, the
semiclassical motion. This condition is presented as follows:

$\left|\frac{\hbar}{2m_{0}\sigma_{x_{0}}}\frac{1-e^{-\gamma
t}}{\gamma}\right|\ll\left|\frac{\overline{p}_{0}}{m_{0}}\frac{1-e^{-\gamma
t}}{\gamma}\right|,$ (55)

or

$\frac{\hbar}{2\sigma_{x_{0}}}\ll\left|\overline{p}_{0}\right|.$ (56)

This result coincides with the condition obtained in (BAG2014, ). For the sake
of simplicity, let us consider $\overline{p}_{0}$ with $\theta_{\xi}=\pi/2$,
see Eqs. (43), such that we can rewrite the above inequality as follows

$\left|\varphi\right|\gg\frac{1}{2}.$ (57)

Given the condition (57), we can identify the SCS as semiclassical states.
Substituting the relation (44), we can write the semiclassical states of the
free particle in the following form

$\displaystyle\Psi_{\overline{x},\overline{p}}^{\sigma_{x}}\left(x,t\right)=\frac{\left(1-\left|\zeta\right|^{2}\right)^{1/4}}{\sqrt{\sqrt{2\pi}e^{r}\left(1-\zeta\right)\sigma_{x_{0}}}}\exp\left[-\frac{1+\zeta}{1-\zeta}\frac{\left(x-\overline{x}\right)^{2}}{4e^{2r}\sigma_{x_{0}}^{2}}+\frac{i\overline{p}_{0}}{2\hbar}\left(2x-\overline{x}\right)+i\phi\right]$
$\displaystyle=\left(\frac{1}{2\pi\sigma_{x}^{2}}\frac{1-\zeta^{\ast}}{1-\zeta}\right)^{1/4}\exp\left[-\frac{\hbar-2i\sigma_{xp}}{\hbar}\frac{\left(x-\overline{x}\right)^{2}}{4\sigma_{x}^{2}}+\frac{i\overline{p}_{0}}{2\hbar}\left(2x-\overline{x}\right)+i\phi\right],$
(58)

where

$\zeta=\frac{4m_{0}\gamma\sigma_{x_{0}}^{2}\sinh\left(r\right)-i\hbar
e^{-r}\left(1-e^{-\gamma
t}\right)}{4m_{0}\gamma\sigma_{x_{0}}^{2}\cosh\left(r\right)+i\hbar
e^{-r}\left(1-e^{-\gamma t}\right)}.$ (59)

We must highlight that the condition (57) is not valid when considering the SS
since the mean values in the position and momentum are null, which justifies
the SS as a class of nonclassical states. On the other hand, the CS brings
special meaning to this condition, which will be investigated in detail below.

###  IV.1 CS of a free particle

Assuming the condition $\zeta=0$, the CS of a free particle takes the form

$\Psi_{\xi,0}\left(x,t\right)=\frac{1}{\sqrt{\sqrt{2\pi}\sigma_{x_{0}}}}\exp\left[-\frac{\left(x+2\sigma_{x_{0}}\xi\right)^{2}}{4\sigma_{x_{0}}^{2}}+\frac{\xi^{2}}{2}-\frac{\left|\xi\right|^{2}}{2}+\frac{i\hbar}{8\gamma
m\sigma_{x_{0}}^{2}}\right].$ (60)

Now, knowing that

$\displaystyle\xi=-\frac{\overline{x}}{2\sigma_{x_{0}}}-\frac{i\sigma_{x_{0}}\overline{p}}{\hbar}=\varphi\exp\left(\frac{i\hbar}{4\sigma_{x_{0}}^{2}m_{0}}\frac{e^{-\gamma
t}-1}{\gamma}\right),\text{ \
}\varphi=\left|\varphi\right|e^{i\theta_{\varphi}},$
$\displaystyle\overline{x}=-2\sigma_{x_{0}}\operatorname{Re}\left(\xi\right)=-2\sigma_{x_{0}}\left|\varphi\right|\cos\left(\theta_{\varphi}+\frac{\hbar}{4m_{0}\sigma_{x_{0}}^{2}}\frac{e^{-\gamma
t}-1}{\gamma}\right),$
$\displaystyle\overline{p}=-\frac{\hbar}{\sigma_{x_{0}}}\operatorname{Im}\left(\xi\right)=-\frac{\hbar}{\sigma_{x_{0}}}\left|\varphi\right|\sin\left(\theta_{\varphi}+\frac{\hbar}{4m_{0}\sigma_{x_{0}}^{2}}\frac{e^{-\gamma
t}-1}{\gamma}\right),$ (61)

we can rewrite the state
$\Psi_{\xi,0}\left(x,t\right)\longrightarrow\Psi_{\overline{x},\overline{p}}^{\sigma_{x_{0}}}\left(x,t\right)$,
as follows

$\Psi_{\overline{x},\overline{p}}^{\sigma_{x_{0}}}\left(x,t\right)=\frac{1}{\sqrt{\sqrt{2\pi}\sigma_{x_{0}}}}\exp\left[-\frac{\left(x-\overline{x}\right)^{2}}{4\sigma_{x_{0}}^{2}}+\frac{i\overline{p}}{2\hbar}\left(2x-\overline{x}\right)+\frac{i\hbar}{8\gamma
m\sigma_{x_{0}}^{2}}\right].$ (62)

In order for the Eqs. (13) to be valid, we must consider the limit (32), which
describes a large mass particle. Also, the displacement parameter must satisfy
the differential Eq. (33). In this case, the mean values present an
oscillatory behavior, indicating that the CS of a free particle with
exponentially time-varying mass mimics the Zitterbewegung effect AWO1990 . It
is important to note that this effect is not classically observed [see eqs.
(45)]. Furthermore, we can interpret that the free particle with the time-
dependent mass brings out the information from when was subjected to the
action of a restoring force with dispersion. This situation can be seen as the
Caldirola-Kanai type oscillator in the limit that the frequency goes to zero,
which describes a free particle with time-dependent mass. Furthermore, these
states minimize the Heisenberg uncertainty relation (47), as can be seen below

$\sigma_{x}\sigma_{p}=\frac{\hbar}{2},\text{ \ }\gamma t\rightarrow\infty.$
(63)

##  V Probabilistic analysis

In this section, we aim to analyze the behavior of the probability density
$\rho_{\overline{x},\overline{p}}^{\sigma_{x}}\left(x,t\right)\equiv\left|\Psi_{\overline{x},\overline{p}}^{\sigma_{x}}\left(x,t\right)\right|^{2}$
and the transition probability
$P_{n}\left(\zeta,\xi\right)\equiv\left|\left\langle
n|\xi,\zeta\right\rangle\right|^{2}$ for the SCS, and we will present their
respective representations for the semiclassical SCS.

###  V.1 Probability density

We start by considering the time-dependent probability density function given
by

$\rho_{\overline{x},\overline{p}}^{\sigma_{x}}\left(x,t\right)=\frac{1}{\sqrt{2\pi}\sigma_{x}}\exp\left[-\frac{\left(x-\overline{x}\right)^{2}}{2\sigma_{x}^{2}}\right],$
(64)

where $\overline{x}$ and $\sigma_{x}$ are given by Eq. (54), and

$\displaystyle\overline{x}_{0}=-2\sigma_{x_{0}}\left|\varphi\right|\cos\left(\theta_{\varphi}\right),\text{
\
}\overline{p}_{0}=-\frac{\hbar\left|\varphi\right|}{\sigma_{x_{0}}}\sin\left(\theta_{\varphi}\right),\text{
\ }\sigma_{x_{0}}=\frac{\hbar}{2\sigma_{p_{0}}}.$ (65)

We see that the Eq. (64) describes a Gaussian wave packet centered in
$x=\overline{x}$ with width $\sigma_{x}$. Note that if we assume $\zeta=0$, we
obtain the CS, while $\varphi=0$, we get the SS. Moreover, taking the limit
$\left|\varphi\right|\gg 1/2$, we will find the condition that allows us to
assume the SCS as being semiclassical states.

(a)

(b)

Figure 3: The probability density, which evolves in time according to Eq.
(64). The dash-dotted curve describes the classical trajectory. We have
assumed $\hbar=1$, $m_{0}=1$, $\sigma_{x_{0}}=1$, and $\gamma=10^{-2}$ with
the initial conditions in (a) $\overline{x}_{0}=0$,
$\overline{p}_{0}=-1/2:\left(\varphi=1/2\text{ and
}\theta_{\varphi}=\pi/2\right)$ and in (b) $\overline{x}_{0}=0$,
$\overline{p}_{0}=-10:\left(\varphi=10\text{ and
}\theta_{\varphi}=\pi/2\right)$ .

The time evolution of the probability density has been shown in Fig. 3. In
Fig. 3a at $t=0$ the probability density is localized around
$\overline{x}_{0}=0$ with a initial width $\sigma_{x_{0}}=1$. As the Figure
shows, with time evolution arises the dispersion of the density due to the
spreading in $x$–direction. In Fig. 3b the probability density plot for a
semiclassical SCS $\left(\varphi\gg 1/2\right)$ has been presented.

###  V.2 Transition probability

Here, we focus on discussing the transition probability. In this case, let us
first admit that the physical system is found in the Fock states and
transitions to another system described by the SCS of a free particle. The
transition probability $P_{n}\left(\zeta,\xi\right)\equiv\left|\left\langle
n|\xi,\zeta\right\rangle\right|^{2}$ is given as follows

$P_{n}\left(\zeta,\xi\right)=\sqrt{1-\left|\zeta\right|^{2}}\exp\left[\frac{\operatorname{Re}\left(\zeta^{\ast}\xi^{2}\right)-\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right]\frac{\left|\zeta\right|^{n}}{2^{n}n!}\left|H_{n}\left(\frac{\xi}{\sqrt{2\zeta}}\right)\right|^{2}.$
(66)

In particular with $\zeta=0$, we get

$P_{n}\left(0,\xi\right)=P_{n}\left(\varphi\right)=\frac{\left|\varphi\right|^{2n}}{n!}\exp\left(-\left|\varphi\right|^{2}\right),$
(67)

that corresponds to the transition probability for the CS.

(a)

(b)

(c)

(d)

Figure 4: The transition probability is shown by considering $\hbar=1$,
$m_{0}=1$, $\sigma_{x_{0}}=1$, $\theta_{\varphi}=\pi/2$, and $\gamma=10^{-2}$.
In (a) and (b) we have kept fixed $\varphi=0$ (SS), and $r=0,2$, respectively.
In (c) has been obtained by assuming $\varphi=4$ (SCS), and $r=0$, while in
(d) we consider $\varphi=4$, and $\zeta=0$ (CS).

As we see by direct comparison between Figs. 4a and 4b, keeping $\varphi$
fixed, the $r$-parameter intensify the transition between even states. On the
other hand, from Fig. 4c, one can see that higher values for
$\varphi$-parameter increase the access to the highest odd states. Moreover,
to higher values of $\varphi\left(\gg 1/2\right)$, we get the representation
of transition probability for semiclassical SCS, i.e., Fig. 4c. Notice that
the assumed value $r=2$ implies a strong squeeze in the coordinate. Finally,
Fig. 4d corresponds to the transition probability for the CS
$\left(\zeta=0\right)$, which is time-independent, as one can see from Eq.
(67).

##  VI Final remarks

In this work, we employ the nonunitary approach to construct the SCS for a
free particle with mass varying exponentially in time. In this approach one
introduces a nonunitary exponential operator composed of a part of the
displacement and squeezed standard operators, in which the squeeze and
displacement parameters are identified with the functions that specify the
integrals of motion. Thus, applying a nonunitary transformation, we related
the integrals of motion directly with the bosonic annihilation operator, which
allowed us to construct the SCS in terms of the time-independent Fock basis.
With this construction, it is straightforward to obtain the condition under
which the SCS satisfies the Schrödinger equation.

We reproduce the usual properties such as nonorthogonality, completeness
relation, and probability density of the SCS. Furthermore, we calculated the
mean values, standard deviation, and uncertainty relations. We obtain a
condition for which the Heisenberg uncertainty relation is minimized on the
initial time and determine a condition on the displacement parameter such that
the SCS can be considered semiclassical states. We show that the class of CS
and SS can be obtained from SCS. In particular, to get the CS, which, in turn,
minimizes the Heisenberg uncertainty relation, it is necessary to impose the
limit of a particle with a large mass, which corresponds to the limit $\gamma
t\rightarrow\infty$. We must have this condition in order to guarantee that
$\hat{A}$ are integrals of motion of the Schrödinger equation. As consequence,
the mean values present an oscillatory behavior, which one can associate with
the Zitterbewegung effect. As for the SS, we show that they can be identified
with the CS of the $SU\left(1,1\right)$ group. Finally, the transition
probability for the SCS, CS, and SS are discussed for several particular
cases.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant No.
165/2018), for partial financial support. ASL and FAB acknowledges support
from CNPq (Grant Nos. 150601/2021-2,312104/2018-9). ASL acknowledges support
from CAPES (Grant No 88887.800922/2023-00). ASP thanks the support of the
Instituto Federal do Pará.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

##  Appendix A Determination of $\Phi$–function

We aim to get the $\Phi$–function present in the states (20). For this, we
must substitute the states $\left|\xi,\zeta\right\rangle$ in the Schrödinger
equation, and multiply it by $\left\langle\zeta,\xi\right|$, to have the
following differential equation for $\Phi$,

$\frac{\dot{\Phi}}{\Phi}=\frac{1}{i\hbar}\frac{\left\langle\xi,\zeta\left|\hat{H}\right|\xi,\zeta\right\rangle}{\left\langle\xi,\zeta|\xi,\zeta\right\rangle}+\dot{\xi}\frac{\left\langle\xi,\zeta\left|\hat{a}^{\dagger}\right|\xi,\zeta\right\rangle}{\left\langle\xi,\zeta|\xi,\zeta\right\rangle}+\frac{1}{2}\dot{\zeta}\frac{\left\langle\xi,\zeta\left|\hat{a}^{\dagger
2}\right|\xi,\zeta\right\rangle}{\left\langle\xi,\zeta|\xi,\zeta\right\rangle}.$
(68)

The derivative of parameters $\zeta$ and $\xi$ is found by assuming the
definitions (15) and (16) together with Eqs. (13), as follows

$\dot{\zeta}=\frac{d}{dt}\left(\frac{g}{f}\right)=-\frac{i\hbar}{2l^{2}m}\left(1+\zeta\right)^{2},\text{
\
}\dot{\xi}=\frac{d}{dt}\left(\frac{\varphi}{f}\right)=-\frac{i\hbar}{2l^{2}m}\left(1+\zeta\right)\xi.$
(69)

The mean value of the operators $\hat{H}\left(t\right)$, $\hat{a}^{\dagger}$
and $\hat{a}^{\dagger 2}$ can be easily obtained by considering these
operators in terms of the integrals of motion (8), in the form

$\displaystyle\hat{a}^{\dagger}=\frac{\hat{B}^{\dagger}-\zeta^{\ast}\hat{B}}{1-\left|\zeta\right|^{2}}+\frac{\zeta^{\ast}\xi-\xi^{\ast}}{1-\left|\zeta\right|^{2}},\text{
\ }\hat{B}\equiv\frac{1}{f}\hat{A},$ $\displaystyle\hat{a}^{\dagger
2}=\frac{\hat{B}^{\dagger 2}+\zeta^{\ast
2}\hat{B}^{2}-2\zeta^{\ast}\hat{B}^{\dagger}\hat{B}+2\left(\zeta^{\ast}\xi-\xi^{\ast}\right)\hat{B}^{\dagger}-2\left(\zeta^{\ast}\xi-\xi^{\ast}\right)\zeta^{\ast}\hat{B}}{\left(1-\left|\zeta\right|^{2}\right)^{2}}+\frac{\left(\zeta^{\ast}\xi-\xi^{\ast}\right)^{2}}{\left(1-\left|\zeta\right|^{2}\right)^{2}}-\frac{\zeta^{\ast}}{1-\left|\zeta\right|^{2}},$
$\displaystyle\hat{H}\left(t\right)=\hbar^{2}\frac{2\left[1+\left|\zeta\right|^{2}+2\operatorname{Re}\left(\zeta\right)\right]\hat{B}^{\dagger}\hat{B}-\left(1+\zeta^{\ast}\right)^{2}\hat{B}^{2}-\left(1+\zeta\right)^{2}\hat{B}^{\dagger
2}}{4l^{2}m\left(1-\left|\zeta\right|^{2}\right)^{2}}$
$\displaystyle+\frac{i\hbar^{2}\operatorname{Im}\left(\zeta\xi^{\ast}-\xi\right)\left[\left(1+\zeta\right)\hat{B}^{\dagger}-\left(1+\zeta^{\ast}\right)\hat{B}\right]}{l^{2}m\left(1-\left|\zeta\right|^{2}\right)^{2}}+\frac{\hbar^{2}\left|\zeta\xi^{\ast}-\xi\right|^{2}}{2l^{2}m\left(1-\left|\zeta\right|^{2}\right)^{2}}$
$\displaystyle+\frac{\hbar^{2}\operatorname{Re}\left[\left(\zeta+\left|\zeta\right|^{2}\right)\left(1-\left|\zeta\right|^{2}\right)-\left(\zeta\xi^{\ast}-\xi\right)^{2}\right]}{2l^{2}m\left(1-\left|\zeta\right|^{2}\right)^{2}}+\frac{\hbar^{2}}{4l^{2}m}.$
(70)

Substituting the relations (70) together with the condition (19), we can
rewrite Eq. (68) as follows

$\frac{\dot{\Phi}}{\Phi}=\frac{i\hbar}{4l^{2}m}\left(\xi^{2}-\zeta-1\right)=\frac{1}{2}\frac{d}{dt}\left(\frac{\zeta^{\ast}\xi^{2}}{1-\left|\zeta\right|^{2}}-\frac{i\hbar}{2l^{2}}\int_{0}^{t}\frac{\zeta+1}{m}d\tau\right).$
(71)

Therefore, the general solution of (68) is given by

$\Phi=C\exp\left(\frac{1}{2}\frac{\zeta^{\ast}\xi^{2}}{1-\left|\zeta\right|^{2}}-\frac{i\hbar}{4l^{2}}\int_{0}^{t}\frac{\zeta+1}{m}d\tau\right),$
(72)

where $C$ is an arbitrary real constant that will be fixed by imposing the
normalization condition on the state $\left|\xi,\zeta\right\rangle$.

##  Appendix B Normalization condition

We will determine the constant $C$ by imposing the normalization of the states
$\left|\xi,\zeta\right\rangle$. First, consider the generating function of the
Hermite polynomials, see the formula $\left(10.13.19\right)$ in (ERD1953, ),

$\displaystyle\exp\left(2yz-z^{2}\right)={\displaystyle\sum\limits_{n=0}^{\infty}}\frac{H_{n}\left(y\right)}{n!}z^{n},$
$\displaystyle z=-\sqrt{\frac{\zeta}{2}}\hat{a}^{\dagger},\text{ \
}y=\frac{\xi}{\sqrt{2\zeta}}.$ (73)

Thus, we can write (20), in the form

$\left|\xi,\zeta\right\rangle=C\exp\left(\frac{1}{2}\frac{\zeta^{\ast}\xi^{2}}{1-\left|\zeta\right|^{2}}-\frac{i\hbar}{4l^{2}}\int_{0}^{t}\frac{\zeta+1}{m}d\tau\right){\displaystyle\sum\limits_{n=0}^{\infty}}\frac{\left(-1\right)^{n}}{\sqrt{n!}}\left(\frac{\zeta}{2}\right)^{\frac{n}{2}}H_{n}\left(\frac{\xi}{\sqrt{2\zeta}}\right)\left|n\right\rangle.$
(74)

In what follows, let us calculate
$\left\langle\zeta,\xi|\xi,\zeta\right\rangle$, which will be given by

$\left\langle\zeta,\xi|\xi,\zeta\right\rangle=\left|C\right|^{2}\exp\left[\frac{\operatorname{Re}\left(\zeta^{\ast}\xi^{2}\right)}{1-\left|\zeta\right|^{2}}+\frac{\hbar}{2l^{2}}\int_{0}^{t}\frac{\operatorname{Im}\left(\zeta\right)}{m}d\tau\right]{\displaystyle\sum\limits_{n=0}^{\infty}}\frac{\left|\zeta\right|^{n}}{2^{n}n!}H_{n}\left(\frac{\xi}{\sqrt{2\zeta}}\right)H_{n}\left(\frac{\xi^{\ast}}{\sqrt{2\zeta^{\ast}}}\right).$
(75)

From the formula (10.13.22) in (ERD1953, ),

$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}H_{n}\left(x\right)H_{n}\left(y\right)\frac{z^{n}}{2^{n}n!}=\frac{1}{\sqrt{1-z^{2}}}\exp\left[\frac{2xyz-\left(x^{2}+y^{2}\right)z^{2}}{1-z^{2}}\right],$
$\displaystyle x=\frac{\xi}{\sqrt{2\zeta}},\text{ \
}y=\frac{\xi^{\ast}}{\sqrt{2\zeta^{\ast}}},\text{ \ }z=\left|\zeta\right|,$
(76)

and imposing the normalization of the states $\left|\xi,\zeta\right\rangle$,
i.e., $\left\langle\zeta,\xi|\xi,\zeta\right\rangle=1$, we have $C$ in the
form

$C=\left(1-\left|\zeta_{0}\right|^{2}\right)^{1/4}\exp\left(-\frac{1}{2}\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right).$
(77)

##  Appendix C Completeness relation

Let us get the completeness relation for the states
$\left|\xi,\zeta\right\rangle$. As we can see from Eqs. (15) and (16), the
displacement and squeeze parameters are related to each other. Thus, we will
obtain the completeness relation for the SCS by sweeping the entire complex
plane of the constant $\varphi=\varphi_{0}$. Therefore, we must have

$\displaystyle\int\left|\varphi,\zeta\right\rangle\left\langle\zeta,\varphi\right|d^{2}\varphi=1,\text{
\ }$ $\displaystyle
d^{2}\varphi=\eta\left(u_{1},u_{2}\right)du_{1}du_{2},\text{ \
}u_{1}=\operatorname{Re}\left(\varphi\right),\text{ \
}u_{2}=\operatorname{Im}\left(\varphi\right).$ (78)

where $\eta\left(u_{1},u_{2}\right)$ is a weight function, which will be fixed
in order to ensure the completeness relation (78).

Consider the real functions $z_{1}$ and $z_{2}$, in the following form

$z_{1}=\frac{\cos\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)u_{1}+\sin\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)u_{2}}{\sqrt{2\left|\zeta\right|}\left|f\right|},\text{
\
}z_{2}=\frac{\cos\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)u_{2}-\sin\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)u_{1}}{\sqrt{2\left|\zeta\right|}\left|f\right|},$
(79)

where we made the substitutions $\zeta=\left|\zeta\right|e^{i\theta_{\zeta}}$
and $f=\left|f\right|e^{i\theta_{f}}$. From here, we can write

$\left(\begin{array}[c]{c}z_{1}\\\
z_{2}\end{array}\right)=Z\left(\begin{array}[c]{c}u_{1}\\\
u_{2}\end{array}\right),\text{ \
}Z=\frac{1}{\sqrt{2\left|\zeta\right|}\left|f\right|}\left(\begin{array}[c]{cc}\cos\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)&\sin\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)\\\
-\sin\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)&\cos\left(\theta_{f}+\frac{1}{2}\theta_{\zeta}\right)\end{array}\right).$
(80)

Thus, the element of integration $d^{2}\varphi$ in the new variables $z_{1}$
and $z_{2}$ takes the form

$d^{2}\varphi=\eta\left(z_{1},z_{2}\right)\det\left(Z^{-1}\right)dz_{1}dz_{2}=2\left|f\right|^{2}\left|\zeta\right|\eta\left(z_{1},z_{2}\right)dz_{1}dz_{2}.$
(81)

Therefore, we can rewrite (78), as follows

$\displaystyle\int\left|\varphi,\zeta\right\rangle\left\langle\zeta,\varphi\right|d^{2}\varphi=2\left|f\right|^{2}\left|\zeta\right|\sqrt{1-\left|\zeta\right|^{2}}{\displaystyle\sum\limits_{n,n^{\prime}=0}^{\infty}}\frac{\left(-1\right)^{n+n^{\prime}}\zeta^{\frac{n}{2}}\left(\zeta^{\ast}\right)^{\frac{n^{\prime}}{2}}}{\sqrt{2^{n+n^{\prime}}n!n^{\prime}!}}\left|n\right\rangle\left\langle
n^{\prime}\right|\times$
$\displaystyle\int\int\exp\left(-\frac{2\left|\zeta\right|}{1+\left|\zeta\right|}z_{1}^{2}-\frac{2\left|\zeta\right|}{1-\left|\zeta\right|}z_{2}^{2}\right)H_{n}\left(z_{1}+iz_{2}\right)H_{n^{\prime}}\left(z_{1}-iz_{2}\right)\eta\left(z_{1},z_{2}\right)dz_{1}dz_{2}.$
(82)

Now $\left(i\right)$ making the following identification

$a=\frac{2\left|\zeta\right|}{1+\left|\zeta\right|},\text{ \
}b=\frac{2\left|\zeta\right|}{1-\left|\zeta\right|},$ (83)

$\left(ii\right)$ considering the orthogonality of the Hermite polynomials
(ISM2015, ; EIJ1990, ),

$\int\int
dxdyH_{n}\left(x+iy\right)H_{n^{\prime}}\left(x-iy\right)e^{-ax^{2}-by^{2}}=\frac{\pi}{\sqrt{ab}}2^{n}n!\left(\frac{a+b}{ab}\right)^{n}\delta_{n,n^{\prime}},$
(84)

and $\left(iii\right)$ choosing the weight function
$\eta\left(z_{1},z_{2}\right)=\left(\pi\mu\right)^{-1}$, we will have

$\int\left|\varphi,\zeta\right\rangle\left\langle\zeta,\varphi\right|d^{2}\varphi={\displaystyle\sum\limits_{n=0}^{\infty}}\left|n\right\rangle\left\langle
n\right|=1.$ (85)

Thus, we show that the resolution of the identity for the SCS of a free
particle is satisfied in the complex plane of the constant $\varphi$.

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