# Exact Solution of a Time-Dependent Quantum Harmonic Oscillator with Two
Frequency Jumps via the Lewis-Riesenfeld Dynamical Invariant Method

Stanley S. Coelho<EMAIL_ADDRESS>Faculdade de Física,
Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil  Lucas Queiroz
<EMAIL_ADDRESS>Faculdade de Física, Universidade Federal do Pará,
66075-110, Belém, Pará, Brazil  Danilo T. Alves<EMAIL_ADDRESS>Faculdade de
Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil  Centro
de Física, Universidade do Minho, P-4710-057, Braga, Portugal

(August 27, 2024)

###### Abstract

Harmonic oscillators with multiple abrupt jumps in their frequencies have been
investigated by several authors during the last decades. We investigate the
dynamics of a quantum harmonic oscillator with initial frequency $\omega_{0}$,
that undergoes a sudden jump to a frequency $\omega_{1}$ and, after a certain
time interval, suddenly returns to its initial frequency. Using the Lewis-
Riesenfeld method of dynamical invariants, we present expressions for the mean
energy value, the mean number of excitations, and the transition
probabilities, considering the initial state different from the fundamental.
We show that the mean energy of the oscillator, after the jumps, is equal or
greater than the one before the jumps, even when $\omega_{1}<\omega_{0}$. We
also show that, for particular values of the time interval between the jumps,
the oscillator returns to the same initial state.

##  I Introduction

The quantum harmonic oscillator potential with time-dependent parameters is
relevant in modeling several problems in physics and has been investigated [1,
2, 3, 4, 5, 6, 7, 8, 9]. For example, the interaction between a spinless
charged quantum particle and a time-dependent external classical
electromagnetic field can be studied through a harmonic potential whose
frequency depends explicitly on time [4, 10, 11, 12, 13], and this is used to
model the quantum motion of this particle in a trap [14, 15, 16, 17, 18, 19].
In the context of quantum electrodynamics, this potential is useful, for
instance, to describe the free electromagnetic field in nonstationary media
[8, 9, 20]. In the context of shortcuts to adiabaticity, time-dependent
quantum oscillators have also been considered [21, 22, 23, 24, 25, 26]. Other
applications are found in relativistic quantum mechanics, quantum field
theory, dynamical Casimir effect and gravitation [27, 28, 29, 30, 31, 32, 33].

A particular case of a quantum harmonic oscillator with time-dependent
parameters, that shows sudden frequency jumps, is investigated, for instance,
in Refs. [34, 35, 36, 37, 21, 24, 23, 38, 39, 40, 41]. Under such jumps (or
any time dependence in the parameters), a classical oscillator in its ground
state remains in the same state, whereas a quantum oscillator can become
excited [35]. Moreover, the wave functions of quantum harmonic oscillators
with time-dependent parameters describe squeezed states [5, 42, 9]. For
example, a sudden change in the oscillation frequency of
${}^{\text{85}}\text{Rb}$ atoms in the vibrational fundamental state of a one-
dimensional optical lattice generates squeezed states [43]. The description of
squeezed states is relevant, for instance, in the implementation of schemes
for noise minimization in quantum sensors, which increases their sensitivity
(see, for instance, Ref. [44] and references therein). Subtle points involving
the squeezed states for the model of two frequency jumps were investigated,
for instance, by Tibaduiza et al. in Ref. [40], where the solution for this
case was obtained via algebraic method.

In the present paper, we investigate the dynamics of a quantum harmonic
oscillator with initial frequency $\omega_{0}$, that undergoes a sudden jump
to a frequency $\omega_{1}$ and, after a certain time interval, suddenly
returns to its initial frequency. Instead of using the algebraic method used
in Ref. [40], here we use the Lewis-Riesenfeld (LR) method of dynamical
invariants. The LR method [2, 3, 4] enables the calculation of the exact wave
function of a system subjected, for instance, to a harmonic oscillator
potential with time-dependent parameters, such as mass and frequency [5, 42,
45]. Using this method, we show that the results for the squeeze parameters,
the quantum fluctuations of the position and momentum operators, and the
probability amplitude of a transition from the fundamental state to an
arbitrary energy eigenstate coincide with those found in Ref. [40]. In
addition, using the same LR method, we also obtain expressions for the mean
energy value and for the mean number of excitations (which were not calculated
in Ref. [40]), and for the transition probabilities considering the initial
state different from the fundamental (which generalizes the formula found in
Ref. [40]).

The paper is organized as follows. In Sec. II.1, we review some results on the
application of the LR method to the quantum harmonic oscillator with time-
dependent frequency. In Sec. II.2, we define the squeezing parameters and,
from these and the oscillator wave function obtained via the LR method, we
determine the quantum fluctuations of the position, momentum and Hamiltonian
operators, the mean number of excitations, and the transition probabilities
between different states. In Sec. III, we apply the results of previous
sections to the model of Ref. [40], and analyze their physical implications.
In Sec. IV, we present our final remarks.

##  II Analytical Method

###  II.1 The Wave Function of the Harmonic Oscillator via Lewis-Riesenfeld
Method

Let us consider the one-dimensional Schrödinger equation for a system whose
Hamiltonian $\hat{H}(t)$ explicitly depends on time [46, 47, 48],

$\displaystyle i\hbar\frac{\partial\Psi(x,t)}{\partial
t}=\hat{H}(t)\Psi(x,t).$ (1)

According to the LR method [2, 3, 4, 5, 42, 9], given an invariant Hermitian
operator $\hat{I}(t)$, which satisfies

$\displaystyle\frac{\partial\hat{I}(t)}{\partial
t}+\frac{1}{i\hbar}\bigl{[}\hat{I}(t),\hat{H}(t)\bigr{]}=0,$ (2)

a particular solution $\Psi_{n}(x,t)$ of Eq. (1) is

$\displaystyle\Psi_{n}(x,t)=\exp\left[i\alpha_{n}(t)\right]\Phi_{n}(x,t),$ (3)

in which $\Phi_{n}(x,t)$ are the eigenfunctions of $\hat{I}(t)$, found from

$\displaystyle\hat{I}(t)\Phi_{n}(x,t)=\lambda_{n}\Phi_{n}(x,t),$ (4)

with $\lambda_{n}$ being time independent eigenvalues of $\hat{I}(t)$, and
$\alpha_{n}(t)$ phase functions, obtained from the equation

$\displaystyle\frac{d\alpha_{n}(t)}{dt}=\int_{-\infty}^{+\infty}dx\,\Phi_{n}^{*}(x,t)\biggl{[}i\frac{\partial}{\partial
t}-\frac{1}{\hbar}\hat{H}(t)\biggr{]}\Phi_{n}(x,t).$ (5)

The general solution $\Psi(x,t)$ of Eq. (1) is

$\Psi(x,t)=\sum_{n=0}^{\infty}C_{n}\Psi_{n}(x,t),$ (6)

where the time independent coefficients $C_{n}$ depend only on the initial
conditions.

Specifically, for a time-dependent one-dimensional harmonic oscillator with
mass $m_{0}$, whose time-dependence is contained purely in its oscillation
frequency $\omega(t)$, the Hamiltonian is given by

$\displaystyle\hat{H}(t)=\frac{\hat{p}^{2}}{2m_{0}}+\frac{1}{2}m_{0}\,\omega(t)^{2}\hat{x}^{2},$
(7)

where $\hat{x}$ and $\hat{p}$ are position and momentum operators,
respectively, with $\left[\hat{x},\hat{p}\right]=i\hbar$. An operator
$\hat{I}(t)$ associated with Eq. (7) is [4, 5, 42, 9]

$\displaystyle\hat{I}(t)=\frac{1}{2}\left\\{\left[\frac{\hat{x}}{\rho(t)}\right]^{2}+\left[\rho(t)\hat{p}-m_{0}\dot{\rho}(t)\hat{x}\right]^{2}\right\\},$
(8)

wherein $\rho(t)$ is a real parameter which is solution of the Ermakov-Pinney
equation [49, 50, 51, 52]

$\displaystyle\ddot{\rho}(t)+\omega(t)^{2}\rho(t)=\frac{1}{m_{0}^{2}\rho(t)^{3}}.$
(9)

The eigenfunctions of $\hat{I}(t)$, given by Eq. (8), are

$\displaystyle\Phi_{n}(x,t)=\frac{1}{\sqrt{2^{n}n!}}\Phi_{0}(x,t)H_{n}\left[\frac{x}{\hbar^{\frac{1}{2}}\rho(t)}\right],$
(10)

where $H_{n}$ are the Hermite polynomials of order $n$ [53] ,
$\lambda_{n}=\left(n+1/2\right)\hbar$, and

$\displaystyle\Phi_{0}(x,t)=\biggl{[}\frac{1}{\pi\hbar\rho(t)^{2}}\biggr{]}^{\frac{1}{4}}\exp\biggl{\\{}\frac{im_{0}}{2\hbar}\biggl{[}\frac{\dot{\rho}(t)}{\rho(t)}+\frac{i}{m_{0}\rho(t)^{2}}\biggr{]}x^{2}\biggr{\\}}.$
(11)

From Eq. (5), the functions $\alpha_{n}(t)$ are given by

$\displaystyle\alpha_{n}(t)=-\frac{1}{m_{0}}\left(n+\frac{1}{2}\right)\int_{0}^{t}\frac{dt^{\prime}}{\rho\left(t^{\prime}\right)^{2}}.$
(12)

Thus, from Eqs. (3), (10) and (12), the wave function $\Psi_{n}(x,t)$
associated with the Hamiltonian (7) is

$\displaystyle\Psi_{n}(x,t)=\frac{1}{\sqrt{2^{n}n!}}\exp\left[-\frac{i}{m_{0}}\left(n+\frac{1}{2}\right)\int_{0}^{t}\frac{dt^{\prime}}{\rho(t^{\prime})^{2}}\right]$
$\displaystyle\times\Phi_{0}(x,t)H_{n}\left[\frac{x}{\hbar^{\frac{1}{2}}\rho(t)}\right].$
(13)

For the case in which the frequency is always constant
[$\omega(t)=\omega_{0}$], the solution of Eq. (9) is $\rho(t)=\rho_{0}$, where
[4, 42, 6]

$\displaystyle\rho_{0}=\frac{1}{\sqrt{m_{0}\omega_{0}}}.$ (14)

Therefore, Eq. (II.1) falls back to the wave function of a harmonic oscillator
with time independent mass and frequency, $\Psi_{n}^{(0)}(x,t)$, given by [46,
47, 48]

$\displaystyle\Psi_{n}^{(0)}(x,t)=\frac{1}{\sqrt{2^{n}n!}}\Bigl{(}\frac{m_{0}\omega_{0}}{\pi\hbar}\Bigr{)}^{\frac{1}{4}}\exp\biggl{[}-i\left(n+\frac{1}{2}\right)\omega_{0}t\biggr{]}$
$\displaystyle\times\exp\biggl{(}-\frac{m_{0}\omega_{0}x^{2}}{2\hbar}\biggr{)}H_{n}\biggl{[}\left(\frac{m_{0}\omega_{0}}{\hbar}\right)^{\frac{1}{2}}x\biggr{]}.$
(15)

###  II.2 Squeeze Parameters, Quantum Fluctuations, Mean Number of
Excitations, and Transition Probability

As discussed in Refs. [54, 5, 42, 9], the quantum states of the time-dependent
oscillator, characterized by the wave function $\Psi_{n}(x,t)$ [Eq. (II.1)],
are squeezed. Thus, we can define the squeeze parameter $r(t)$ and the squeeze
phase $\phi(t)$, which specify the squeezed state, in terms of the parameter
$\rho(t)$ [55]:

$\displaystyle
r(t)=\cosh^{-1}\Bigl{\\{}\left(4m_{0}\omega_{0}\right)^{-\frac{1}{2}}\bigl{[}m_{0}^{2}\dot{\rho}(t)^{2}+\rho(t)^{-2}$
$\displaystyle+2m_{0}\omega_{0}+m_{0}^{2}\omega_{0}^{2}\rho(t)^{2}\bigr{]}^{\frac{1}{2}}\Bigr{\\}},$
(16)
$\displaystyle\phi(t)=\cos^{-1}\left\\{\frac{1+m_{0}\omega_{0}\rho(t)^{2}-2\cosh^{2}[r(t)]}{2\sinh[r(t)]\cosh[r(t)]}\right\\},$
(17)

with $r(t)\geq 0$ and $0\leq\phi(t)\leq 2\pi$. From Eq. (II.1), one can also
obtain the expected value of a given observable $\hat{O}(t)$ in the state
$\Psi_{n}(x,t)$, as

$\displaystyle\langle\hat{O}(t)\rangle(n,t)=\int_{-\infty}^{+\infty}dx\,\Psi_{n}^{*}(x,t)\,\hat{O}(t)\,\Psi_{n}(x,t),$
(18)

which, from Eqs. (16) and (17), can be written in terms of $r(t)$ and
$\phi(t)$. For the operators $\hat{x}$ and $\hat{p}$, one has [56]:

$\displaystyle\langle\hat{x}\rangle(n,t)=\langle\hat{p}\rangle(n,t)=0,$ (19)
$\displaystyle\langle\hat{x}^{2}\rangle(n,t)=\left(n+\frac{1}{2}\right)\frac{\hbar}{m_{0}\omega_{0}}\bigl{\\{}\cosh^{2}\left[r(t)\right]+\sinh^{2}\left[r(t)\right]$
$\displaystyle+\,2\sinh\left[r(t)\right]\cosh\left[r(t)\right]\cos\left[\phi(t)\right]\bigr{\\}},$
(20)
$\displaystyle\langle\hat{p}^{2}\rangle(n,t)=\left(n+\frac{1}{2}\right)m_{0}\omega_{0}\hbar\bigl{\\{}\cosh^{2}\left[r(t)\right]+\sinh^{2}\left[r(t)\right]$
$\displaystyle-\,2\sinh\left[r(t)\right]\cosh\left[r(t)\right]\cos\left[\phi(t)\right]\bigr{\\}},$
(21)

where it follows, from Eqs. (7), (20), and (21), that

$\displaystyle\langle\hat{H}(t)\rangle(n,t)=\frac{\langle\hat{p}^{2}\rangle(n,t)}{2m_{0}}+\frac{1}{2}m_{0}\,\omega(t)^{2}\langle\hat{x}^{2}\rangle(n,t).$
(22)

From Eqs. (19)-(21), one finds the variances of the operators $\hat{x}$:

$\displaystyle\langle[\Delta\hat{x}]^{2}\rangle(n,t)=\langle\hat{x}^{2}\rangle(n,t)-[\langle\hat{x}\rangle(n,t)]^{2},$
(23)

and $\hat{p}$:

$\displaystyle\langle[\Delta\hat{p}]^{2}\rangle(n,t)=\langle\hat{p}^{2}\rangle(n,t)-[\langle\hat{p}\rangle(n,t)]^{2},$
(24)

which implies the uncertainty relationship

$\displaystyle\langle[\Delta\hat{x}]^{2}\rangle(n,t)\langle[\Delta\hat{p}]^{2}\rangle(n,t)\geq\biggl{(}n+\frac{1}{2}\biggr{)}^{2}\hbar^{2}\bigl{\\{}\cosh^{4}[r(t)]$
$\displaystyle+\sinh^{4}[r(t)]-2\sinh^{2}[r(t)]\cosh^{2}[r(t)]\cos[2\phi(t)]\bigr{\\}}.$
(25)

Due to the time-dependence of the frequency, one can also determine the mean
number of excitations $\langle\hat{N}\rangle(n,t)$ that a system, subjected to
this potential, can undergo. This is given by [57, 58, 59]

$\displaystyle\langle\hat{N}\rangle(n,t)=n+\left(2n+1\right)\sinh^{2}\left[r(t)\right].$
(26)

For the fundamental state $n=0$, one finds
$\langle\hat{N}\rangle(0,t)=\sinh^{2}\left[r(t)\right]$, a result that agrees
with Refs. [56, 33] for vacuum squeezed states. This means that, a system,
even in the fundamental state, could be excited due to the temporal variations
in its frequency. The system subjected to the time-dependent harmonic
potential can also make transitions between different states, since time-
dependent potentials induce quantum systems to make transitions [47, 46, 48].
Let us consider that the system is initially at a stationary state
$\Psi_{m}^{(0)}(x,t=0)$ with frequency $\omega_{0}$ and, due to a modification
in its frequency from $\omega_{0}$ to $\omega(t)$, it evolves to a new state
$\Psi_{m}(x,t)$ [Eq. (II.1)]. In this way, the probability to find the system
in the state $\Psi_{n}^{(0)}(x,t)$ [Eq. (15)], is given by [47, 46, 48]

$\displaystyle{\cal P}(t)_{m\to
n}=\biggl{|}\int_{-\infty}^{+\infty}dx\,\Psi_{n}^{*(0)}(x,t)\Psi_{m}(x,t)\biggr{|}^{2}.$
(27)

Using Eqs. (II.1) and (15) one can find that ${\cal P}(t)_{m\to n}=0$ for odd
values of $|n-m|$, and [60, 57]

$\displaystyle{\cal P}(t)_{m\to
n}=\frac{2^{m+n}\min(m,n)!^{2}\left\\{\sinh[r(t)]\right\\}^{|n-m|}}{m!n!\cosh[r(t)]}$
$\displaystyle\times\left[\sum_{k=\frac{|n-m|}{2}}^{\frac{n+m}{2}}\frac{\left(\begin{array}[]{c}\frac{n+m}{2}\\\
k\end{array}\right)\left(\begin{array}[]{c}\frac{n+m+2k-2}{4}\\\
\frac{n+m}{2}\end{array}\right)k!}{\left(k-\frac{|n-m|}{2}\right)!\cosh^{k}[r(t)]}\right]^{2},$
(32)

for even values of $|n-m|$, where $\min(m,n)$ is the smallest value between
$m$ and $n$. Note that, the fact that Eq. (32) is non-zero only for $|n-m|$
even is related to the parity of the harmonic potential [61]. From Eqs. (26)
(making $n=0$) and (32), we can relate the probability ${\cal P}(t)_{m\to n}$
to the mean number of excitations in the fundamental state
$\langle\hat{N}\rangle(0,t)$:

$\displaystyle{\cal P}(t)_{m\to
n}=\frac{2^{m+n}\min(m,n)!^{2}\bigl{[}\langle\hat{N}\rangle(0,t)\bigr{]}^{\frac{|n-m|}{2}}}{m!n!\bigl{[}\langle\hat{N}\rangle(0,t)+1\bigr{]}^{\frac{1}{2}}}$
$\displaystyle\times\left[\sum_{k=\frac{|n-m|}{2}}^{\frac{n+m}{2}}\frac{\left(\begin{array}[]{c}\frac{n+m}{2}\\\
k\end{array}\right)\left(\begin{array}[]{c}\frac{n+m+2k-2}{4}\\\
\frac{n+m}{2}\end{array}\right)k!}{\left(k-\frac{|n-m|}{2}\right)!\bigl{[}\langle\hat{N}\rangle(0,t)+1\bigr{]}^{\frac{k}{2}}}\right]^{2}.$
(37)

It follows that if the mean number of excitations in the fundamental state is
non-zero, then the oscillator will have non-zero probabilities of making
transitions between different energy levels. When we consider $m=n=0$ in Eq.
(32) [or in Eq. (37)], one has the probability of persistence in the
fundamental state, and, as a consequence, one can also obtain the probability
of excitation, given by $1-{\cal P}(t)_{0\,\to\,0}$ [40].

##  III Oscillator with two frequency jumps

Now, we apply the formulas shown in Sec. II to investigate the model discussed
in Ref. [40], namely an oscillator with

$\displaystyle\omega(t)=\begin{cases}\omega_{0},&t<0,\\\
\omega_{1},&0<t<\tau,\\\ \omega_{0},&t>\tau,\end{cases}$ (38)

in which $\omega_{0}$ and $\omega_{1}$ are constant frequencies, and $\tau$ is
the length of the time interval between the frequency jumps.

###  III.1 Solution and General Behavior of the $\rho(t)$ Parameter

Due to the form of Eq. (38), the $\rho(t)$ parameter can be written as

$\displaystyle\rho(t)=\begin{cases}\rho_{0},&t<0,\\\ \rho_{1}(t),&0<t<\tau,\\\
\rho_{2}(t),&t>\tau,\end{cases}$ (39)

where $\rho_{0}$ is given in Eq. (14), and $\rho_{1}(t)$ and $\rho_{2}(t)$ are
calculated next.

####  III.1.1 Interval $0<t<\tau$

For the interval $0<t<\tau$, the equation to be solved is

$\displaystyle\ddot{\rho}_{1}(t)+\omega_{1}^{2}\rho_{1}(t)=\frac{1}{m_{0}^{2}\rho_{1}(t)^{3}},$
(40)

with the conditions [6]

$\displaystyle\rho_{1}\left(t=0\right)=\frac{1}{\sqrt{m_{0}\omega_{0}}},\;\;\dot{\rho}_{1}\left(t=0\right)=0.$
(41)

The general solution for $\rho_{1}(t)$ is of the form [50, 51]

$\displaystyle\rho_{1}(t)=\bigl{[}A_{1}\sin^{2}\left(\omega_{1}t\right)+2C_{1}\sin\left(\omega_{1}t\right)\cos\left(\omega_{1}t\right)$
$\displaystyle+B_{1}\cos^{2}\left(\omega_{1}t\right)\bigr{]}^{\frac{1}{2}},$
(42)

and the relationship between the constants $A_{1}$, $B_{1}$ and $C_{1}$ is

$\displaystyle A_{1}B_{1}-C_{1}^{2}=\frac{1}{m_{0}^{2}\omega_{1}^{2}}.$ (43)

Then, applying conditions (41) to Eq. (42) and using relation (43), we get

$\displaystyle\rho_{1}(t)=\left[\frac{\omega_{0}\sin^{2}\left(\omega_{1}t\right)}{m_{0}\omega_{1}^{2}}+\frac{\cos^{2}\left(\omega_{1}t\right)}{m_{0}\omega_{0}}\right]^{\frac{1}{2}}.$
(44)

####  III.1.2 Interval $t>\tau$

In the interval $t>\tau$, the Ermakov-Pinney equation has the form

$\displaystyle\ddot{\rho}_{2}(t)+\omega_{0}^{2}\rho_{2}(t)=\frac{1}{m_{0}^{2}\rho_{2}(t)^{3}}.$
(45)

The general solution of Eq. (45) is

$\displaystyle\rho_{2}(t)=\bigl{[}A_{2}\sin^{2}\left(\omega_{0}t\right)+2C_{2}\sin\left(\omega_{0}t\right)\cos\left(\omega_{0}t\right)$
$\displaystyle+B_{2}\cos^{2}\left(\omega_{0}t\right)\bigr{]}^{\frac{1}{2}},$
(46)

with the constants $A_{2}$, $B_{2}$ and $C_{2}$ determined from the
relationship

$\displaystyle A_{2}B_{2}-C_{2}^{2}=\frac{1}{m_{0}^{2}\omega_{0}^{2}},$ (47)

and the conditions for continuity

$\displaystyle\rho_{1}(t=\tau)=\rho_{2}(t=\tau),\;\;\;\dot{\rho}_{1}(t=\tau)=\dot{\rho}_{2}(t=\tau).$
(48)

Using Eqs. (44), (46), (47) and (48), the results in

$\displaystyle
A_{2}=\frac{1}{m_{0}\omega_{0}^{3}\omega_{1}^{2}}\Bigl{\\{}\omega_{0}^{2}\omega_{1}^{2}+\bigl{[}(\omega_{0}^{4}-\omega_{1}^{4})\sin^{2}(\omega_{0}\tau)-\omega_{0}^{2}\omega_{1}^{2}$
$\displaystyle+\omega_{1}^{4}\bigr{]}\sin^{2}(\omega_{1}\tau)+2\omega_{0}\omega_{1}(\omega_{0}-\omega_{1})(\omega_{0}+\omega_{1})\sin(\omega_{0}\tau)$
$\displaystyle\times\cos(\omega_{0}\tau)\sin(\omega_{1}\tau)\cos(\omega_{1}\tau)\Bigr{\\}},$
(49) $\displaystyle
B_{2}=\frac{1}{m_{0}\omega_{0}^{3}\omega_{1}^{2}}\Bigl{\\{}\omega_{0}^{2}\omega_{1}^{2}+\bigl{[}\left(\omega_{1}^{4}-\omega_{0}^{4}\right)\sin^{2}\left(\omega_{0}\tau\right)-\omega_{0}^{2}\omega_{1}^{2}$
$\displaystyle+\omega_{0}^{4}\bigr{]}\sin^{2}\left(\omega_{1}\tau\right)-2\omega_{0}\omega_{1}\left(\omega_{0}-\omega_{1}\right)\left(\omega_{0}+\omega_{1}\right)\sin\left(\omega_{0}\tau\right)$
$\displaystyle\times\cos\left(\omega_{0}\tau\right)\sin\left(\omega_{1}\tau\right)\cos\left(\omega_{1}\tau\right)\Bigr{\\}},$
(50) $\displaystyle
C_{2}=\frac{1}{m_{0}\omega_{0}^{3}\omega_{1}^{2}}\Bigl{\\{}\bigl{[}\left(\omega_{0}^{2}+\omega_{1}^{2}\right)\sin\left(\omega_{0}\tau\right)\cos\left(\omega_{0}\tau\right)\sin\left(\omega_{1}\tau\right)$
$\displaystyle-2\omega_{0}\omega_{1}\left(\sin^{2}\left(\omega_{0}\tau\right)-1/2\right)\cos\left(\omega_{1}\tau\right)\bigr{]}\left(\omega_{0}-\omega_{1}\right)$
$\displaystyle\times\left(\omega_{0}+\omega_{1}\right)\sin\left(\omega_{1}\tau\right)\Bigr{\\}}.$
(51)

####  III.1.3 General Behavior

The general solution for the $\rho(t)$ parameter is given by Eq. (39), with
$\rho_{0}$, $\rho_{1}(t)$ and $\rho_{2}(t)$ given by Eqs. (14), (44), (46),
(49), (50), and (51). From these equations, it can be seen that the $\rho(t)$
parameter is a periodic function of time. Moreover, even when the frequency
returns to its initial value $\omega_{0}$, this parameter will still, in
general, be a periodic function of time. However, if we define
$\tau_{u}=u\pi/\omega_{1}$ ($u>0$), and make $\tau=\tau_{l}$, where
$l\in\mathbb{N}$, the $\rho(t)$ parameter returns to $\rho_{0}$ [Eq. (14)],
which means that although the oscillator feels the effect of the change in its
frequency when it jumps from $\omega_{0}$ to $\omega_{1}$, if the frequency
returns to $\omega_{0}$ at $\tau=\tau_{l}$, for $t>\tau_{l}$ the oscillator
behaves as if nothing happened. In other words, if $\tau=\tau_{l}$, the abrupt
change in the frequency is imperceptible to the oscillator when $t>\tau_{l}$.
On the other hand, when $\tau=\tau_{l+1/2}$, the $\rho(t)$ parameter reaches
its maximum value. The behavior of $\rho(t)$ is shown in Fig. 1.

Figure 1: General behavior of $\rho(t)$ as a function of $\omega_{0}t$, with
$\omega_{1}=3\omega_{0}$ and different values of $\tau$ (we consider, for
simplicity, $m_{0}=\omega_{0}=1$ and $\hbar=1$ in arbitrary units). For
$\tau=[\tau_{l+1/2}]_{l=2}=5\pi/6$, we note that the amplitude of oscillation
for the $\rho(t)$ parameter is maximum. For $\tau=59\pi/62$, the amplitude of
oscillation is intermediate. Finally, when $\tau=[\tau_{l}]_{l=3}=\pi$, there
is no oscillation at all, and it follows that $\rho(t)$ becomes time
independent.

###  III.2 Squeeze Parameters

Because the frequency varies abruptly, squeezing occurs in the system [40, 35,
36]. Thus, now, we calculate the parameters $r(t)$ and $\phi(t)$ associated
with the model in Eq. (38). We show that our results for these parameters
agree with those found in Ref. [40] via an exact algebraic method.

####  III.2.1 Parameter $r(t)$

The parameter $r(t)$ for any time interval is given by

$\displaystyle r(t)=\begin{cases}0,&t<0,\\\ r_{1}(t),&0<t<\tau,\\\
r_{2}(t),&t>\tau.\end{cases}$ (52)

Note that $r(t<0)=0$ because the frequency of the oscillator is time
independent in this interval. Using Eq. (44) in Eq. (16) we obtain, for the
interval $0<t<\tau$, the squeezing parameter $r_{1}(t)$, where

$\displaystyle
r_{1}(t)=\cosh^{-1}\Biggl{\\{}\sqrt{1+\biggl{(}\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}\omega_{1}}\biggr{)}^{2}\sin^{2}(\omega_{1}t)}\Biggr{\\}},$
(53)

which is a result that agrees with the one found in Ref. [40]. For the
interval $t>\tau$, using Eq. (46) in Eq. (16), we find that $r_{2}(t)$, with

$\displaystyle r_{2}(t)=r_{1}(\tau),$ (54)

that also agrees with Ref. [40]. Note that for $\tau=\tau_{l}$, one has
$r_{2}(t)=0$. In Fig. 2 (also found in Ref. [40]), one can see the behavior of
$r(t)$ for some values of $\tau$.

Figure 2: Behavior of the squeeze parameter $r(t)$ as a function of
$\omega_{0}t$, where $\omega_{1}=3\omega_{0}$ (we consider $\omega_{0}=1$ in
arbitrary units).

####  III.2.2 Parameter $\phi(t)$

The squeeze phase $\phi(t)$ for any time interval has the form

$\displaystyle\phi(t)=\begin{cases}\text{undefined},&t<0,\\\
\phi_{1}(t),&0<t<\tau,\\\ \phi_{2}(t),&t>\tau,\end{cases}$ (55)

where $\phi(t)$ for $t<0$ is undefined because there is no squeeze in this
interval. Using Eqs. (44) and (53) in Eq. (17), we obtain that the squeeze
phase for the interval $0<t<\tau$, $\phi_{1}(t)$, is given by

$\displaystyle\phi_{1}(t)=\cos^{-1}\Bigl{\\{}\left(\omega_{0}^{4}-\omega_{1}^{4}\right)\sin^{2}(\omega_{1}t)\bigl{[}\bigl{(}4\omega_{0}^{2}\omega_{1}^{2}+\bigl{(}\omega_{1}^{2}$
$\displaystyle-\omega_{0}^{2}\bigr{)}^{2}\sin^{2}(\omega_{1}t)\bigr{)}\left(\omega_{1}^{2}-\omega_{0}^{2}\right)^{2}\sin^{2}(\omega_{1}t)\bigr{]}^{-\frac{1}{2}}\Bigr{\\}},$
(56)

which also agrees with Ref. [40]. To calculate the squeeze phase in the
interval $t>\tau$, the reasoning is analogous, simply substituting Eqs. (46)
and (54) into Eq. (17).

Figure 3: Behavior of the squeeze phase $\phi(t)$ as a function of
$\omega_{0}t$, where $\omega_{1}=3\omega_{0}$ (we consider $\omega_{0}=1$ in
arbitrary units).

From Fig. 3, it can be seen that the squeezing phase will continue to vary in
time even for $t>\tau$. Due of this time dependence, the fluctuations of the
$\hat{x}$ and $\hat{p}$ operators will continue to depend on time in this
interval, as we will see later in Sec. III.3 [see Eqs. (23) and (24)]. Another
point to be observed, in Fig. 3, concerns the behavior of the squeezing phase
in the interval $t>\tau$, when $\tau=\tau_{l}$ (in the specific case of Fig.
3, $\tau_{l}=\pi$). Since the squeeze parameter $r_{2}(t)$ [Eq. (54)] is zero
in this case, the system is no longer squeezed. Consequently, the squeeze
phase is undefined for $\tau_{l}$. Therefore, its effect on the system will be
negligible because there will be no more squeezing.

###  III.3 Quantum Fluctuations

The variance of the $\hat{x}$ operator for any time interval is given by

$\displaystyle\langle[\Delta\hat{x}]^{2}\rangle(n,t)=\begin{cases}\langle[\Delta\hat{x}]_{0}^{2}\rangle(n),&t<0,\\\
\langle[\Delta\hat{x}]_{1}^{2}\rangle(n,t),&0<t<\tau,\\\
\langle[\Delta\hat{x}]_{2}^{2}\rangle(n,t),&t>\tau,\end{cases}$ (57)

where [46]

$\displaystyle\langle[\Delta\hat{x}]_{0}^{2}\rangle(n)=\left(n+\frac{1}{2}\right)\frac{\hbar}{m_{0}\omega_{0}}.$
(58)

Substituting Eqs. (53) and (56) into (23) we find, for the $\hat{x}$ operator
in the interval $0<t<\tau$,

$\displaystyle\langle[\Delta\hat{x}]_{1}^{2}\rangle(n,t)=\left[\frac{\omega_{0}^{2}}{\omega_{1}^{2}}\sin^{2}(\omega_{1}t)+\cos^{2}(\omega_{1}t)\right]\langle[\Delta\hat{x}]_{0}^{2}\rangle(n),$
(59)

which is in agreement with Refs. [40, 35]. For the interval $t>\tau$, the
procedure is analogous, using Eqs. (16), (17) and (46) in Eq. (23). In Fig. 4,
we show the behavior of $\langle[\Delta\hat{x}]^{2}\rangle(n,t)$ for $n=0$.

Figure 4: Behavior of the $\langle[\Delta\hat{x}]^{2}\rangle(n,t)$ for $n=0$
as a function of $\omega_{0}t$, where $\omega_{1}=3\omega_{0}$ (we consider,
for simplicity, $m_{0}=\omega_{0}=1$ and $\hbar=1$ in arbitrary units).

Similarly, for the variance of the $\hat{p}$ operator, we have

$\displaystyle\langle[\Delta\hat{p}]^{2}\rangle(n,t)=\begin{cases}\langle[\Delta\hat{p}]_{0}^{2}\rangle(n),&t<0,\\\
\langle[\Delta\hat{p}]_{1}^{2}\rangle(n,t),&0<t<\tau,\\\
\langle[\Delta\hat{p}]_{2}^{2}\rangle(n,t),&t>\tau,\end{cases}$ (60)

being [46]

$\displaystyle\langle[\Delta\hat{p}]_{0}^{2}\rangle(n)=\left(n+\frac{1}{2}\right)\hbar
m_{0}\omega_{0}.$ (61)

Through Eqs. (53), (56) and (24), we find, for the interval $0<t<\tau$, the
expression [40, 35]

$\displaystyle\langle[\Delta\hat{p}]_{1}^{2}\rangle(n,t)=\left[\frac{\omega_{1}^{2}}{\omega_{0}^{2}}\sin^{2}(\omega_{1}t)+\cos^{2}(\omega_{1}t)\right]\langle[\Delta\hat{p}]_{0}^{2}\rangle(n).$
(62)

To calculate the variance of the operator $\hat{p}$ in the interval $t>\tau$,
we must use Eqs. (16), (17) and (46) in Eq. (24). The general behavior of
$\langle[\Delta\hat{p}]^{2}\rangle(n,t)$, when $n=0$, is schematized in Fig.
5.

Figure 5: Behavior of the $\langle[\Delta\hat{p}]^{2}\rangle(n,t)$ for $n=0$
as a function of $\omega_{0}t$, where $\omega_{1}=3\omega_{0}$ (we consider,
for simplicity, $m_{0}=\omega_{0}=1$ and $\hbar=1$ in arbitrary units).

Thus, it is direct to see that the uncertainty relation between these
operators in the interval $0<t<\tau$ has the form

$\displaystyle\langle[\Delta\hat{x}]_{1}^{2}\rangle(n,t)\langle[\Delta\hat{p}]_{1}^{2}\rangle(n,t)\geq\Biggl{\\{}1+\biggl{[}\biggl{(}\frac{\omega_{1}^{2}-\omega_{0}^{2}}{\omega_{0}\omega_{1}}\biggr{)}$
$\displaystyle\times\sin(\omega_{1}t)\cos(\omega_{1}t)\biggr{]}^{2}\Biggr{\\}}\langle[\Delta\hat{x}]_{0}^{2}\rangle(n)\langle[\Delta\hat{p}]_{0}^{2}\rangle(n),$
(63)

and the uncertainty relation for the interval $t>\tau$ is obtained in a
similar way. Clearly, when $\omega_{1}=\omega_{0}$, the uncertainty relation
[Eq. (63)] falls back to the uncertainty relation of a time independent
oscillator [46]. Furthermore, the uncertainty relation for an oscillator with
time independent frequency is also reobtained when $\tau=\tau_{l}$, as shown
in Fig. 6.

Figure 6: Behavior of the
$\langle[\Delta\hat{x}]^{2}\rangle(n,t)\langle[\Delta\hat{p}]^{2}\rangle(n,t)$
for $n=0$ as a function of $\omega_{0}t$, where $\omega_{1}=3\omega_{0}$ (we
consider, for simplicity, $m_{0}=\omega_{0}=1$ and $\hbar=1$ in arbitrary
units).

The results found by us (via the LR method) given in Eqs. (53), (54), (56),
(59), and (62) are in agreement with those found in Ref. [40]. Hereafter, we
use the LR method to obtain new results concerning the model given in Eq.
(38).

###  III.4 Mean Energy

The expected value of the Hamiltonian operator is identified as the mean
energy of the system, that is,
$E\left(n,t\right)=\langle\hat{H}(t)\rangle(n,t)$. The mean energy for the
model in Eq. (38) can be written as

$\displaystyle E(n,t)=\begin{cases}E_{0}(n),&t<0,\\\ E_{1}(n,t),&0<t<\tau,\\\
E_{2}(n,t),&t>\tau,\end{cases}$ (64)

wherein [46]

$\displaystyle E_{0}(n)=\left(n+\frac{1}{2}\right)\hbar\omega_{0}.$ (65)

For the interval $0<t<\tau$, through Eqs. (59), (62), (65) and (22), we find
that the mean energy is time independent, given by

$\displaystyle
E_{1}(n,t)=\frac{1}{2}\left(1+\frac{\omega_{1}^{2}}{\omega_{0}^{2}}\right)E_{0}(n).$
(66)

When $\omega_{1}=\omega_{0}$, Eq. (66) reduces to $E_{1}(n,t)=E_{0}(n)$, as
expected. Note that for $\omega_{1}/\omega_{0}<1$, $E_{1}(n,t)<E_{0}(n)$,
whereas for $\omega_{1}/\omega_{0}>1$, we have $E_{1}(n,t)>E_{0}(n)$. When
$\omega_{1}=0$, which means that the system is free in interval $0<t<\tau$, we
have $E_{1}(n,t)=E_{0}(n)/2$. This can also be obtained by making
$\omega_{0}\gg\omega_{1}$, which leads to $E_{1}(n,t)\approx E_{0}(n)/2$.

For the interval $t>\tau$, from Eqs. (16), (17), (22), (46), and (65), we have
that the mean energy $E_{2}(n,t)$ is given by

$\displaystyle
E_{2}(n,t)=\biggl{[}1+\frac{1}{2}\biggl{(}\frac{\omega_{1}^{2}-\omega_{0}^{2}}{\omega_{0}\omega_{1}}\biggr{)}^{2}\sin^{2}(\omega_{1}\tau)\biggr{]}E_{0}(n).$
(67)

Note that Eq. (67) is independent of $t$, and $E_{2}(n,t)\geq E_{0}(n)$, even
for $\omega_{1}<\omega_{0}$. The behavior of the ratio $E_{2}(n,t)/E_{0}(n)$
is illustrated in Fig. 7.

Figure 7: Ratio $E_{2}(n,t)/E_{0}(n)$ as a function of $\omega_{0}\tau$ and
$\omega_{1}/\omega_{0}$. The dashed line corresponds to
$\omega_{1}=\omega_{0}$. The dotted lines correspond to $\tau=\tau_{l}$. The
dot-dashed lines correspond to $\tau=\tau_{l+1/2}$.

For $\omega_{1}=\omega_{0}$ (dashed line in Fig. 7), Eq. (67) recovers
$E_{2}(n,t)=E_{0}(n)$, as expected. Furthermore, for $\tau=\tau_{l}$ (dotted
lines in Fig. 7), Eq. (67) also gives $E_{2}(n,t)=E_{0}(n)$. In particular,
the energy of this system is maximized when $\tau=\tau_{l+1/2}$ (dot-dashed
lines in Fig. 7). For $\omega_{1}/\omega_{0}\to 0$, we obtain
$E_{2}(n,t)=\left(1+\omega_{0}^{2}\tau^{2}/2\right)E_{0}(n)$. Moreover, from
Eqs. (54) and (67), we obtain

$\displaystyle
E_{2}(n,t)=\left\\{2\cosh^{2}\left[r_{2}(t)\right]-1\right\\}E_{0}(n).$ (68)

Thus, while there is squeeze, $E_{2}(n,t)>E_{0}(n)$. Therefore, the squeezing
caused by the frequency jumps results in an increase in the mean energy of the
oscillator, with respect to its initial energy $E_{0}(n)$.

It is interesting to investigate the behavior of $E_{2}(n,t)/E_{1}(n,t)$.
Unlike the ratio $E_{2}(n,t)/E_{0}(n)$, which is such that
$E_{2}(n,t)/E_{0}(n)\geq 1$, the ratio $E_{2}(n,t)/E_{1}(n,t)$ can be lesser
than, equal to, or greater than one, as shown in Fig. 8. More specifically,
for $\omega_{1}/\omega_{0}<1$, we have $E_{2}(n,t)/E_{1}(n,t)>1$, and when
$\omega_{1}/\omega_{0}\to 0$, we find
$E_{2}(n,t)=\left(2+\omega_{0}^{2}\tau^{2}\right)E_{1}(n,t)$. For
$\omega_{1}/\omega_{0}>1$, the ratio $E_{2}(n,t)/E_{1}(n,t)$ oscillates
between zero and one (see Fig. 8). We highlight that, besides the trivial case
$\omega_{1}=\omega_{0}$, there are other values of the ratio
$\omega_{1}/\omega_{0}$ that result in $E_{2}(n,t)=E_{1}(n,t)$.

Figure 8: Ratio $E_{2}(n,t)/E_{1}(n,t)$ as a function of
$\omega_{1}/\omega_{0}$. For $\omega_{1}/\omega_{0}<1$,
$E_{2}(n,t)/E_{1}(n,t)>1$, whereas for $\omega_{1}/\omega_{0}>1$,
$E_{2}(n,t)/E_{1}(n,t)\leq 1$.

###  III.5 Mean Number of Excitations

The mean number of excitations, $\langle\hat{N}\rangle(n,t)$, for the model in
Eq. (38), is given by

$\displaystyle\langle\hat{N}\rangle(n,t)=\begin{cases}\langle\hat{N}\rangle_{0}(n),&t<0,\\\
\langle\hat{N}\rangle_{1}(n,t),&0<t<\tau,\\\
\langle\hat{N}\rangle_{2}(n,t),&t>\tau,\end{cases}$ (69)

where [46]

$\displaystyle\langle\hat{N}\rangle_{0}(n)=n.$ (70)

Given this, for the interval $0<t<\tau$, by means of Eqs. (26) and (53), we
have

$\displaystyle\langle\hat{N}\rangle_{1}(n,t)=n+\left(n+\frac{1}{2}\right)\biggl{[}\frac{1}{2}\biggl{(}\frac{\omega_{1}^{2}-\omega_{0}^{2}}{\omega_{0}\omega_{1}}\biggr{)}^{2}\sin^{2}(\omega_{1}t)\biggr{]}.$
(71)

We remark the time dependence in $\langle\hat{N}\rangle_{1}(n,t)$, whereas the
mean energy in the interval $0<t<\tau$ [Eq. (66)] is time independent.

For the interval $t>\tau$, through Eqs. (26) and (54), we obtain
$\langle\hat{N}\rangle_{2}(n,t)$, given by

$\displaystyle\langle\hat{N}\rangle_{2}(n,t)=\langle\hat{N}\rangle_{1}(n,\tau).$
(72)

We also remark that for $\tau=\tau_{l}$, the behavior of the system returns to
that of the time independent oscillator found before the frequency jumps, i.e,
$\langle\hat{N}\rangle_{2}(n,t)=\langle\hat{N}\rangle_{0}(n)$. We highlight
that there is excitation even for $n=0$, which means that, under jumps in its
frequency, a quantum oscillator initially in its ground state can become
excited (a classical oscillator in its ground state would remain in the same
state). In addition, excitation can also occur when $\omega_{1}/\omega_{0}<1$,
as shown in Fig. 9.

Figure 9: Some examples of mean number of excitations
$\langle\hat{N}\rangle_{2}(0,t)$ that an oscillator could undergo as a
function of $\omega_{1}/\omega_{0}$ for different values of $\omega_{0}\tau$.

Using Eqs. (67) and (72), we can also write $E_{2}(n,t)$ as

$\displaystyle
E_{2}(n,t)=\left[\langle\hat{N}\rangle_{2}(n,t)+\frac{1}{2}\right]\hbar\omega_{0},$
(73)

which has the same structure as the expression for the energy eigenvalues of a
time independent oscillator [see Eq. (65)]. The time evolution of
$\langle\hat{N}\rangle(0,t)$, given by Eq. (69) for $n=0$, is shown in Fig.
10.

Figure 10: Mean number of excitations $\langle\hat{N}\rangle(0,t)$ as a
function of $\omega_{0}t$ for different values of $\tau$, where
$\omega_{1}=3\omega_{0}$ (we consider $\omega_{0}=1$ in arbitrary units).

###  III.6 Transition Probability

The general transition probability, ${\cal P}(t)_{m\to n}$, for the model in
Eq. (38), is given by

$\displaystyle{\cal P}(t)_{m\to n}=\begin{cases}\delta_{m,n},&t<0,\\\ {\cal
P}_{1}(t)_{m\to n},&0<t<\tau,\\\ {\cal P}_{2}(t)_{m\to n},&t>\tau,\end{cases}$
(74)

with $\delta_{m,n}$ being the Kronecker delta. Using Eqs. (32) and (53) [or
Eqs. (37) and (71) with $n=0$], we find, for the interval $0<t<\tau$, ${\cal
P}_{1}(t)_{m\to n}$, whose expression is

$\displaystyle{\cal P}_{1}(t)_{m\to
n}=\frac{2^{m+n}\min(m,n)!^{2}\Bigl{[}\Bigl{(}\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}\omega_{1}}\Bigr{)}\sin(\omega_{1}t)\Bigr{]}^{|n-m|}}{m!n!\Bigl{[}1+\Bigl{(}\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}\omega_{1}}\Bigr{)}^{2}\sin^{2}(\omega_{1}t)\Bigr{]}^{\frac{1}{2}}}$
$\displaystyle\times\left[\sum_{k=\frac{|n-m|}{2}}^{\frac{n+m}{2}}\frac{\left(\begin{array}[]{c}\frac{n+m}{2}\\\
k\end{array}\right)\left(\begin{array}[]{c}\frac{n+m+2k-2}{4}\\\
\frac{n+m}{2}\end{array}\right)k!}{\left(k-\frac{|n-m|}{2}\right)!\Bigl{[}1+\Bigl{(}\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}\omega_{1}}\Bigr{)}^{2}\sin^{2}(\omega_{1}t)\Bigr{]}^{\frac{k}{2}}}\right]^{2},$
(79) (80)

for even values of $|n-m|$, and ${\cal P}(t)_{m\to n}=0$ for odd values of
$|n-m|$. Since the $r_{1}(t)$ parameter [Eq. (53)], in this interval, is
explicitly time-dependent, consequently, the transition probability $m\to n$
will also depend.

For the interval $t>\tau$, using Eq. (54) in Eq. (32) [or Eqs. (37) and (72)
with $n=0$], we see that the result is

$\displaystyle{\cal P}_{2}(t)_{m\to n}={\cal P}_{1}(\tau)_{m\to n}.$ (81)

Thus, even when the frequency returns to $\omega_{0}$ after an instant $\tau$,
one can find a non-zero $m\to n$ transition probability, depending on the
value of $\tau$. It is noticeable, from Eq. (80), that ${\cal P}(t)_{m\to
n}={\cal P}(t)_{n\to m}$, with such symmetry being a consequence of the parity
of the potential in Eq. (7) [61]. The behavior of ${\cal P}(t)_{1\to n}$ is
illustrated in Figs. 11 (for $\tau=3\tau_{1}/2$) and 12 (for $\tau=\tau_{1}$).
In Fig. 11, as $n$ increases, ${\cal P}(t)_{1\to n}$ decreases. We highlight
that for $t>\tau=\tau_{1}$ in Fig. 12, ${\cal P}(t)_{1\to 1}=1$ or, in other
words, the oscillator remains in its same initial state.

Figure 11: Behavior of ${\cal P}(t)_{1\to n}$ as a function of $\omega_{0}t$,
with $\omega_{1}=2\omega_{0}$ and $\tau=3\tau_{1}/2=3\pi/4$ (we consider
$\omega_{0}=1$ in arbitrary units). Figure 12: Behavior of ${\cal P}(t)_{1\to
n}$ as a function of $\omega_{0}t$, with $\omega_{1}=2\omega_{0}$ and
$\tau=\tau_{1}=\pi/2$ (we consider $\omega_{0}=1$ in arbitrary units).

In addition, for $m=0$, Eq. (81) gives

$\displaystyle{\cal
P}_{2}(t)_{0\,\to\,n}=\frac{n!\left(\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}\omega_{1}}\right)^{n}\sin^{n}(\omega_{1}\tau)}{2^{n}\left(\frac{n}{2}!\right)^{2}\left[1+\left(\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}\omega_{1}}\right)^{2}\sin^{2}(\omega_{1}\tau)\right]^{\frac{n+1}{2}}},$
(82)

where $n=0,2,4,\ldots,$ recovering one of the results found in Ref. [40].
Thus, Eq. (81) generalizes the result for the transition probability found in
Ref. [40]. By also making $n=0$ in Eq. (82), we find the probability of the
oscillator of persisting in the fundamental state, and the probability of the
oscillator being excited, after the frequency returns to $\omega_{0}$, which
is given by $1-{\cal P}_{2}(t)_{0\,\to\,0}$.

##  IV Final Remarks

Using the Lewis-Riesenfeld method, we investigated the dynamics of a quantum
harmonic oscillator that undergoes two abrupt jumps in its frequency [Eq.
(38)]. We reobtained the analytical formulas of Ref. [40] for the squeeze
parameters [Eqs. (53), (54), and (56)], the quantum fluctuations of the
position [Eq. (59)] and momentum [Eq. (62)] operators, and the probability
amplitude of a transition from the fundamental state to an arbitrary energy
eigenstate [Eq. (82)]. We also obtained expressions for the mean energy value
[Eqs. (66) and (67)] and for the mean number of excitations [Eqs. (71) and
(72)] (which were not calculated in Ref. [40]) , and for the transition
probabilities considering the initial state different from the fundamental
[Eqs. (80) and (81)] (which generalizes the formula found in Ref. [40]).

We found that, as expected, the mean energy of the system is independent of
time in each one of the intervals: $t<0$, $0<t<\tau$, and $t>\tau$. Moreover,
we showed that the mean energy of the oscillator after the jumps is equal or
greater than that before these jumps, even when $\omega_{1}<\omega_{0}$. We
also obtained, for $t>\tau\neq\tau_{l}$, a non-null value for the mean number
of excitation when the oscillator starts in the fundamental state [Eqs. (66)
and (67) with $n=0$], which means that, under the jumps in its frequency, a
quantum oscillator, initially in the ground state, can become excited. We
showed that transitions between arbitrary $m$ and $n$ states only occur if
$|n-m|$ is an even number. We highlighted that, for $t>\tau\neq\tau_{l}$ and a
fixed value of $m$, as $n$ increases, ${\cal P}(t)_{m\to n}$ decreases.
Finally, we showed that, for $t>\tau=\tau_{l}$, ${\cal P}(t)_{m\to
n}=\delta_{m,n}$, so that the oscillator returns to the same initial state
(this generalizes, for any initial state $m$, the result found in Ref. [40]
for $m=0$).

###### Acknowledgements.

The authors thank Alexandre Costa and Edson Nogueira for valuable discussions,
as well as Adolfo del Campo, Bogdan M. Mihalcea, Daniel Tibaduiza, and Viktor
Dodonov for their valuable suggestions to this paper. S.S.C. was partialy
supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico -
Brazil (CNPq) by the program PIBIC/CNPq through the project No. 144456/2020-6,
Fundação Amazônia de Amparo a Estudos e Pesquisas (Fapespa) by the program
PIBIC/Fapespa, and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
- Brazil (CAPES), Finance Code 001. L.Q. was also supported by the CAPES,
Finance Code 001.

## References

  * Husimi [1953] Husimi, K.  Miscellanea in Elementary Quantum Mechanics, II.  Prog. Theor. Phys. 1953, 9, 381–402.  https://doi.org/10.1143/ptp/9.4.381.
  * Lewis [1967] Lewis, H.R.  Classical and Quantum Systems with Time-Dependent Harmonic-Oscillator-Type Hamiltonians.  Phys. Rev. Lett. 1967, 18, 510–512.  https://doi.org/10.1103/PhysRevLett.18.636.2.
  * Lewis [1968] Lewis, H.R.  Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators.  J. Math. Phys. 1968, 9, 1976–1986.  https://doi.org/10.1063/1.1664532.
  * Lewis and Riesenfeld [1969] Lewis, H.R.; Riesenfeld, W.B.  An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field.  J. Math. Phys. 1969, 10, 1458–1473.  https://doi.org/10.1063/1.1664991.
  * Pedrosa [1997] Pedrosa, I.A.  Exact wave functions of a harmonic oscillator with time-dependent mass and frequency.  Phys. Rev. A 1997, 55, 3219–3221.  https://doi.org/10.1103/PhysRevA.55.3219.
  * Ciftja [1999] Ciftja, O.  A simple derivation of the exact wavefunction of a harmonic oscillator with time-dependent mass and frequency.  J. Phys. A. Math. Gen. 1999, 32, 6385–6389.  https://doi.org/10.1088/0305-4470/32/36/303.
  * Guasti and Moya-Cessa [2003] Guasti, M.F.; Moya-Cessa, H.  Solution of the Schrödinger equation for time-dependent 1D harmonic oscillators using the orthogonal functions invariant.  J. Phys. A. Math. Gen. 2003, 36, 2069–2076.  https://doi.org/10.1088/0305-4470/36/8/305.
  * Pedrosa and Rosas [2009] Pedrosa, I.A.; Rosas, A.  Electromagnetic Field Quantization in Time-Dependent Linear Media.  Phys. Rev. Lett. 2009, 103, 010402.  https://doi.org/10.1103/PhysRevLett.103.010402.
  * Pedrosa [2011] Pedrosa, I.A.  Quantum electromagnetic waves in nonstationary linear media.  Phys. Rev. A 2011, 83, 032108.  https://doi.org/10.1103/PhysRevA.83.032108.
  * Dodonov et al. [1994] Dodonov, V.; Man’ko, V.; Polynkin, P.  Geometrical squeezed states of a charged particle in a time-dependent magnetic field.  Phys. Lett. A 1994, 188, 232–238.  https://doi.org/10.1016/0375-9601(94)90444-8.
  * Xiu-wei et al. [1999] Xiu-wei, X.; Ting-qi, R.; Sheng-dian, L.  Analytic solution for one-dimensional quantum oscillator with a variable frequency.  Acta Phys. Sin. 1999, 8, 641–645.  https://doi.org/10.1088/1004-423X/8/9/001.
  * Aguiar and Guedes [2016] Aguiar, V.; Guedes, I.  Entropy and information of a spinless charged particle in time-varying magnetic fields.  J. Math. Phys. 2016, 57, 092103.  https://doi.org/10.1063/1.4962923.
  * Dodonov and Horovits [2018] Dodonov, V.V.; Horovits, M.B.  Squeezing of Relative and Center-of-Orbit Coordinates of a Charged Particle by Step-Wise Variations of a Uniform Magnetic Field with an Arbitrary Linear Vector Potential.  J. Russ. Laser Res. 2018, 39, 389–400.  https://doi.org/10.1007/s10946-018-9733-1.
  * Brown [1991] Brown, L.S.  Quantum motion in a Paul trap.  Phys. Rev. Lett. 1991, 66, 527–529.  https://doi.org/10.1103/PhysRevLett.66.527.
  * Agarwal and Kumar [1991] Agarwal, G.S.; Kumar, S.A.  Exact quantum-statistical dynamics of an oscillator with time-dependent frequency and generation of nonclassical states.  Phys. Rev. Lett. 1991, 67, 3665–3668.  https://doi.org/10.1103/PhysRevLett.67.3665.
  * Mihalcea [2009] Mihalcea, B.M.  A quantum parametric oscillator in a radiofrequency trap.  Phys. Scr. 2009, T135, 014006.  https://doi.org/10.1088/0031-8949/2009/T135/014006.
  * Aguiar et al. [2016] Aguiar, V.; Nascimento, J.; Guedes, I.  Exact wave functions and uncertainties for a spinless charged particle in a time-dependent Penning trap.  Int. J. Mass Spectrom. 2016, 409, 21–28.  https://doi.org/10.1016/j.ijms.2016.09.007.
  * Menicucci and Milburn [2007] Menicucci, N.C.; Milburn, G.J.  Single trapped ion as a time-dependent harmonic oscillator.  Phys. Rev. A 2007, 76, 052105.  https://doi.org/10.1103/PhysRevA.76.052105.
  * Pedrosa [2021] Pedrosa, I.A.  On the Quantization of the London Superconductor.  Braz. J. Phys. 2021, 51, 401–405.  https://doi.org/10.1007/s13538-020-00851-x.
  * Choi [2010] Choi, J.R.  Interpreting quantum states of electromagnetic field in time-dependent linear media.  Phys. Rev. A 2010, 82, 055803.  https://doi.org/10.1103/PhysRevA.82.055803.
  * Salamon et al. [2009] Salamon, P.; Hoffmann, K.H.; Rezek, Y.; Kosloff, R.  Maximum work in minimum time from a conservative quantum system.  Phys. Chem. Chem. Phys. 2009, 11, 1027–1032.  https://doi.org/10.1039/B816102J.
  * Schaff et al. [2010] Schaff, J.F.; Song, X.L.; Vignolo, P.; Labeyrie, G.  Fast optimal transition between two equilibrium states.  Phys. Rev. A 2010, 82, 033430.  https://doi.org/10.1103/PhysRevA.82.033430.
  * Chen et al. [2010] Chen, X.; Ruschhaupt, A.; Schmidt, S.; del Campo, A.; Guéry-Odelin, D.; Muga, J.G.  Fast Optimal Frictionless Atom Cooling in Harmonic Traps: Shortcut to Adiabaticity.  Phys. Rev. Lett. 2010, 104, 063002.  https://doi.org/10.1103/PhysRevLett.104.063002.
  * Stefanatos et al. [2010] Stefanatos, D.; Ruths, J.; Li, J.S.  Frictionless atom cooling in harmonic traps: A time-optimal approach.  Phys. Rev. A 2010, 82, 063422.  https://doi.org/10.1103/PhysRevA.82.063422.
  * Dupays et al. [2021] Dupays, L.; Spierings, D.C.; Steinberg, A.M.; del Campo, A.  Delta-kick cooling, time-optimal control of scale-invariant dynamics, and shortcuts to adiabaticity assisted by kicks.  Phys. Rev. Res. 2021, 3, 033261.  https://doi.org/10.1103/PhysRevResearch.3.033261.
  * Martínez-Tibaduiza et al. [2021] Martínez-Tibaduiza, D.; Pires, L.; Farina, C.  Time-dependent quantum harmonic oscillator: a continuous route from adiabatic to sudden changes.  J. Phys. B At. Mol. Opt. Phys. 2021, 54, 205401.  https://doi.org/10.1088/1361-6455/ac36ba.
  * Landim and Guedes [2000] Landim, R.R.; Guedes, I.  Wave functions for a Dirac particle in a time-dependent potential.  Phys. Rev. A 2000, 61, 054101.  https://doi.org/10.1103/PhysRevA.61.054101.
  * Gao et al. [1998] Gao, X.C.; Fu, J.; Li, X.H.; Gao, J.  Invariant formulation and exact solutions for the relativistic charged Klein-Gordon field in a time-dependent spatially homogeneous electric field.  Phys. Rev. A 1998, 57, 753–761.  https://doi.org/10.1103/PhysRevA.57.753.
  * Dodonov and Klimov [1996] Dodonov, V.V.; Klimov, A.B.  Generation and detection of photons in a cavity with a resonantly oscillating boundary.  Phys. Rev. A 1996, 53, 2664–2682.  https://doi.org/10.1103/PhysRevA.53.2664.
  * Dodonov et al. [1990] Dodonov, V.; Klimov, A.; Man’ko, V.  Generation of squeezed states in a resonator with a moving wall.  Phys. Lett. A 1990, 149, 225–228.  https://doi.org/10.1016/0375-9601(90)90333-J.
  * Pedrosa and Guedes [2004] Pedrosa, I.A.; Guedes, I.  Exact quantum states of an inverted pendulum under time-dependent gravitation.  Int. J. Mod. Phys. A 2004, 19, 4165–4172.  https://doi.org/10.1142/S0217751X04019731.
  * Carvalho et al. [2004] Carvalho, A.M.d.M.; Furtado, C.; Pedrosa, I.A.  Scalar fields and exact invariants in a Friedmann-Robertson-Walker spacetime.  Phys. Rev. D 2004, 70, 123523.  https://doi.org/10.1103/PhysRevD.70.123523.
  * Greenwood [2015] Greenwood, E.  Time-dependent particle production and particle number in cosmological de Sitter space.  Int. J. Mod. Phys. D 2015, 24, 1550031.  https://doi.org/10.1142/S0218271815500315.
  * Janszky and Yushin [1986] Janszky, J.; Yushin, Y.  Squeezing via frequency jump.  Opt. Commun. 1986, 59, 151–154.  https://doi.org/10.1016/0030-4018(86)90468-2.
  * Janszky and Adam [1992] Janszky, J.; Adam, P.  Strong squeezing by repeated frequency jumps.  Phys. Rev. A 1992, 46, 6091–6092.  https://doi.org/10.1103/PhysRevA.46.6091.
  * Kiss et al. [1994] Kiss, T.; Janszky, J.; Adam, P.  Time evolution of harmonic oscillators with time-dependent parameters: A step-function approximation.  Phys. Rev. A 1994, 49, 4935–4942.  https://doi.org/10.1103/PhysRevA.49.4935.
  * Moya-Cessa and Fernández Guasti [2003] Moya-Cessa, H.; Fernández Guasti, M.  Coherent states for the time dependent harmonic oscillator: the step function.  Phys. Lett. A 2003, 311, 1–5.  https://doi.org/10.1016/S0375-9601(03)00461-4.
  * Stefanatos [2017a] Stefanatos, D.  Minimum-Time Transitions between Thermal and Fixed Average Energy States of the Quantum Parametric Oscillator.  SIAM J. Control Optim. 2017, 55, 1429–1451.  https://doi.org/10.1137/16M1088697.
  * Stefanatos [2017b] Stefanatos, D.  Minimum-Time Transitions Between Thermal Equilibrium States of the Quantum Parametric Oscillator.  IEEE Trans. Automat. Contr. 2017, 62, 4290–4297.  https://doi.org/10.1109/TAC.2017.2684083.
  * Tibaduiza et al. [2020a] Tibaduiza, D.M.; Pires, L.; Szilard, D.; Zarro, C.A.D.; Farina, C.; Rego, A.L.C.  A Time-Dependent Harmonic Oscillator with Two Frequency Jumps: an Exact Algebraic Solution.  Braz. J. Phys. 2020, 50, 634–646.  https://doi.org/10.1007/s13538-020-00770-x.
  * Tibaduiza et al. [2020b] Tibaduiza, D.M.; Pires, L.; Rego, A.L.C.; Szilard, D.; Zarro, C.; Farina, C.  Efficient algebraic solution for a time-dependent quantum harmonic oscillator.  Phys. Scr. 2020, 95, 105102.  https://doi.org/10.1088/1402-4896/abb254.
  * Pedrosa et al. [1997] Pedrosa, I.A.; Serra, G.P.; Guedes, I.  Wave functions of a time-dependent harmonic oscillator with and without a singular perturbation.  Phys. Rev. A 1997, 56, 4300–4303.  https://doi.org/10.1103/PhysRevA.56.4300.
  * Xin et al. [2021] Xin, M.; Leong, W.S.; Chen, Z.; Wang, Y.; Lan, S.Y.  Rapid Quantum Squeezing by Jumping the Harmonic Oscillator Frequency.  Phys. Rev. Lett. 2021, 127, 183602.  https://doi.org/10.1103/PhysRevLett.127.183602.
  * Wolf et al. [2019] Wolf, F.; Shi, C.; Heip, J.C.; Gessner, M.; Pezzè, L.; Smerzi, A.; Schulte, M.; Hammerer, K.; Schmidt, P.O.  Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions.  Nat. Commun. 2019, 10, 2929.  https://doi.org/10.1038/s41467-019-10576-4.
  * Choi [2004] Choi, J.R.  The dependency on the squeezing parameter for the uncertainty relation in the squeezed states of the time-dependent oscillator.  Int. J. Mod. Phys. B 2004, 18, 2307–2324.  https://doi.org/10.1142/S0217979204026135.
  * Sakurai and Napolitano [2020] Sakurai, J.J.; Napolitano, J.  Modern Quantum Mechanics, 3º ed.; Cambridge University Press: Cambridge, UK, 2020; pp. 83–88.
  * Griffiths [2018] Griffiths, D.J.  Introduction to Quantum Mechanics, 3º ed.; Cambridge University Press: Cambridge, UK, 2018; pp. 39–55.
  * Cohen-Tannoudji et al. [2019] Cohen-Tannoudji, C.; Diu, B.; Laloe, F.  Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications, 2º ed.; Wiley-VCH: Weinheim, Germany, 2019; pp. 502–521.
  * Prykarpatskyy [2018] Prykarpatskyy, Y.  Steen–Ermakov–Pinney Equation and Integrable Nonlinear Deformation of the One-Dimensional Dirac Equation.  J. Math. Sci. 2018, 231, 820–826.  https://doi.org/10.1007/s10958-018-3851-8.
  * Pinney [1950] Pinney, E.  The nonlinear differential equation $y^{\prime\prime}+p(x)y+cy^{-3}=0$.  Proc. Am. Math. Soc. 1950, 1, 681.  https://doi.org/10.1090/S0002-9939-1950-0037979-4.
  * de Lima et al. [2009] de Lima, A.L.; Rosas, A.; Pedrosa, I.  Quantum dynamics of a particle trapped by oscillating fields.  J. Mod. Opt. 2009, 56, 75–80.  https://doi.org/10.1080/09500340802495834.
  * Cariñena and de Lucas [2009] Cariñena, J.F.; de Lucas, J.  Applications of Lie systems in dissipative Milne-Pinney equations.  Int. J. Geom. Methods Mod. Phys. 2009, 06, 683–699.  https://doi.org/10.1142/S0219887809003758.
  * Weber and Arfken [2003] Weber, H.J.; Arfken, G.B.  Essential mathematical methods for physicists, 6º ed.; Academic Press: San Diego, USA, 2003; pp. 638–642.
  * Pedrosa [1987] Pedrosa, I.A.  Comment on “Coherent states for the time-dependent harmonic oscillator”.  Phys. Rev. D 1987, 36, 1279–1280.  https://doi.org/10.1103/PhysRevD.36.1279.
  * Daneshmand and Tavassoly [2017] Daneshmand, R.; Tavassoly, M.K.  Dynamics of Nonclassicality of Time- and Conductivity-Dependent Squeezed States and Excited Even/Odd Coherent States.  Commun. Theor. Phys. 2017, 67, 365–376.  https://doi.org/10.1088/0253-6102/67/4/365.
  * Guerry and Knight [2005] Guerry, C.C.; Knight, P.L.  Introductory Quantum Optics, 1º ed.; Cambridge University Press: Cambridge, UK, 2005; pp. 150–165.
  * Kim et al. [1989] Kim, M.S.; de Oliveira, F.A.M.; Knight, P.L.  Properties of squeezed number states and squeezed thermal states.  Phys. Rev. A 1989, 40, 2494–2503.  https://doi.org/10.1103/PhysRevA.40.2494.
  * Marian [1992] Marian, P.  Higher-order squeezing and photon statistics for squeezed thermal states.  Phys. Rev. A 1992, 45, 2044–2051.  https://doi.org/10.1103/PhysRevA.45.2044.
  * Moeckel and Kehrein [2009] Moeckel, M.; Kehrein, S.  Real-time evolution for weak interaction quenches in quantum systems.  Ann. Phys. (N. Y). 2009, 324, 2146–2178.  https://doi.org/10.1016/j.aop.2009.03.009.
  * Kim et al. [1989] Kim, M.; de Oliveira, F.; Knight, P.  Photon number distributions for squeezed number states and squeezed thermal states.  Opt. Commun. 1989, 72, 99–103.  https://doi.org/10.1016/0030-4018(89)90263-0.
  * Popov and Perelomov [1969] Popov, V.; Perelomov, A.  Parametric Excitation of a Quantum Oscillator.  Sov. J. Exp. Theor. Phys. 1969, 30, 1375–1390.