Time-independent model
We are interested in a family of QHOs including all possible position and momentum combinations up to second order49,
$$\begin{aligned} {\hat{H}} = \frac{ 1 }{ 2m } {\hat{p}}^{2} + \frac{ 1 }{ 2 } m \omega ^{2} {\hat{x}}^{2} + \alpha _{x} {\hat{x}} + \alpha _{p} {\hat{p}} + \alpha _{xp} \{ {\hat{x}} , {\hat{p}} \} + \alpha _{0}, \end{aligned}$$
(2)
where the mass is given by m, the oscillator frequency by \(\omega\), the external driving of the position by \(\alpha _{x}\) with units of \(\text{energy} \cdot \text{position}^{-1}\), the constant shift of the momentum by \(\alpha _{p}\) with units of \(\text{energy} \cdot \text{momentum}^{-1}\), the external scaling constant or parametric drive \(\alpha _{xp}\) with units of \(\text{time}^{-1}\) and a constant energy \(\alpha _{0}\). We use the standard notation for the anti-commutator \(\{{\hat{x}},{\hat{p}}\} = {\hat{x}} {\hat{p}} + {\hat{p}} {\hat{x}}\).
The Hamiltonian model shows a closed underlying algebra,
$$\begin{aligned} \begin{aligned} \left[ {\hat{x}},{\hat{p}}\right]&=i\hbar ,&\left[ \{{\hat{x}},{\hat{p}}\},{\hat{x}}\right]&=-2i\hbar {\hat{x}},\\ \left[ {\hat{x}},{\hat{p}}^{2}\right]&=2i\hbar {\hat{p}},&\left[ \{{\hat{x}},{\hat{p}}\},{\hat{p}}\right]&=2i\hbar {\hat{p}},\\ \left[ {\hat{p}},{\hat{x}}^{2}\right]&=-2i\hbar {\hat{x}},&\left[ \{{\hat{x}},{\hat{p}}\},{\hat{x}}^{2}\right]&=-4i\hbar {\hat{x}}^{2}, \\ \left[ {\hat{x}}^{2},{\hat{p}}^{2}\right]&=2i\hbar \{{\hat{x}},{\hat{p}}\},&\left[ \{{\hat{x}},{\hat{p}}\},{\hat{p}}^{2}\right]&=4i\hbar {\hat{p}}^{2}, \\ \end{aligned} \end{aligned}$$
(3)
that help us define a group with three types of unitary transformations. First, a translation,
$$\begin{aligned} \begin{aligned} e^{ \frac{i}{\hbar } \left[ \beta _{p} {\hat{x}} + \beta _{x} {\hat{p}} \right] } \, {\hat{x}} \, e^{ – \frac{i}{\hbar } \left[ \beta _{p} {\hat{x}} + \beta _{x} {\hat{p}} \right] }&= {\hat{x}} + \beta _{x}, \\ e^{ \frac{i}{\hbar } \left[ \beta _{p} {\hat{x}} + \beta _{x} {\hat{p}} \right] } \, {\hat{p}} \, e^{ – \frac{i}{\hbar } \left[ \beta _{p} {\hat{x}} + \beta _{x} {\hat{p}} {\hat{x}}\right] }&= {\hat{p}} – \beta _{p}, \end{aligned} \end{aligned}$$
(4)
where the real constant displacement parameters \(\beta _{x}\) and \(\beta _{p}\) have units of \(\text{position}\) and \(\text{momentum}\), respectively. A rotation,
$$\begin{aligned} \begin{aligned} e^{\frac{i}{\hbar } \left[ \theta _{x}^{2} {\hat{x}}^{2} + \theta _{p}^{2} {\hat{p}}^{2} \right] } \, {\hat{x}} \, e^{-\frac{i}{\hbar }\left[ \theta _{x}^{2} {\hat{x}}^{2} + \theta _{p}^{2} {\hat{p}}^{2} \right] }&= \cos {\left( 2 \theta _{x} \theta _p \right) } \, {\hat{x}} + \frac{ \theta _{p}}{\theta _{x} } \sin {\left( 2 \theta _{x} \theta _p \right) } \, {\hat{p}}, \\ e^{\frac{i}{\hbar }\left[ \theta _{x}^{2} {\hat{x}}^{2} + \theta _{p}^{2} {\hat{p}}^{2}\right] } \, {\hat{p}} \, e^{-\frac{i}{\hbar }\left[ \theta _{x}^{2} {\hat{x}}^{2} + \theta _{p}^{2} {\hat{p}}^{2}\right] }&= \cos {\left( 2 \theta _{x} \theta _p \right) } \, {\hat{p}} – \frac{\theta _{x}}{ \theta _{p}} \sin {\left( 2 \theta _{x} \theta _p \right) } \, {\hat{x}}, \end{aligned} \end{aligned}$$
(5)
where the rotation parameters \(\theta _{x}^{2}\) and \(\theta _{p}^{2}\) have units of \(\text{momentum} \cdot \text{position}^{-1}\) and \(\text{position} \cdot \text{momentum}^{-1}\), respectively. It will be helpful later to note that these rotations take a simpler form,
$$\begin{aligned} \begin{aligned} \lim _{\theta _{x} \rightarrow 0} e^{\frac{i}{\hbar }\left[ \theta _{x}^{2} {\hat{x}}^{2} + \theta _{p}^{2} {\hat{p}}^{2}\right] } \, {\hat{x}} \, e^{-\frac{i}{\hbar }\left[ \theta _{x} {\hat{x}}^{2} + \theta _{p} {\hat{p}}^{2}\right] }&= {\hat{x}} + 2 \theta _{p}^{2} \, {\hat{p}}, \\ \lim _{\theta _{p} \rightarrow 0} e^{\frac{i}{\hbar }\left[ \theta _{x}^{2} {\hat{x}}^{2} + \theta _{p}^{2} {\hat{p}}^{2}\right] } \, {\hat{p}} \, e^{-\frac{i}{\hbar }\left[ \theta _{x}^{2} {\hat{x}}^{2} + \theta _{p}^{2} {\hat{p}}^{2}\right] }&= {\hat{p}} – 2 \theta _{x}^{2} \, {\hat{x}}, \end{aligned} \end{aligned}$$
(6)
when one of the parameters is zero. Finally, we have a squeezing transformation,
$$\begin{aligned} \begin{aligned} e^{ \frac{ i }{ \hbar } \beta \{{\hat{x}},{\hat{p}}\} } \, {\hat{x}} \, e^{ – \frac{ i }{ \hbar } \beta \{ {\hat{x}} , {\hat{p}} \} }&= {\hat{x}} e^{2 \beta }, \\ e^{ \frac{ i }{ \hbar } \beta \{ {\hat{x}} , {\hat{p}} \} } \, {\hat{p}} \, e^{ – \frac{ i }{ \hbar } \beta \{ {\hat{x}} , {\hat{p}} \} }&= {\hat{p}} e^{ -2 \beta }, \end{aligned} \end{aligned}$$
(7)
that provides inverse scaling for position and momentum with a dimensionless real scaling parameter \(\beta\).
In order to create insight, we use these transformations to diagonalize our Hamiltonian in Eq. (2). First, we move into a reference frame,
$$\begin{aligned} \vert {\Psi } \rangle = e^{-\frac{i}{\hbar } \alpha _{0} t} \vert {\psi _{1}} \rangle , \end{aligned}$$
(8)
rotating at the constant frequency \(\alpha _{0}\) to obtain the effective Hamiltonian,
$$\begin{aligned} \hat{H_{1}} = {\hat{H}} – \alpha _{0}, \end{aligned}$$
(9)
without the constant bias. In order to deal with the forced position and momentum terms proportional to \({\hat{x}}\) and \({\hat{p}}\), we apply the translation,
$$\begin{aligned} \vert {\psi _{1}} \rangle = e^{-\frac{i}{\hbar } \left( \beta _{p} {\hat{x}} + \beta _{x} {\hat{p}} \right) } \vert {\psi _{2}} \rangle , \end{aligned}$$
(10)
with parameters,
$$\begin{aligned} \begin{aligned} \beta _{x}&= \frac{ 2 m \alpha _{p} \alpha _{xp} – \alpha _{x} }{m(\omega ^{2} – 4 \alpha _{xp}^{2}) } , \\ \beta _{p}&= m\alpha _{p} – \frac{2\alpha _{xp}(\alpha _{x}-2 m \alpha _{p}\alpha _{xp})}{\omega ^2-4\alpha _{xp}^2} , \end{aligned} \end{aligned}$$
(11)
that yields an effective Hamiltonian,
$$\begin{aligned} \hat{H_{2}} = \frac{1}{2m} {\hat{p}}^{2} + \frac{1}{2} m \omega ^{2} {\hat{x}}^{2} + \alpha _{xp} \{{\hat{x}},{\hat{p}}\} + l, \end{aligned}$$
(12)
where the accumulation constant,
$$\begin{aligned} l = \frac{1}{2m}\beta _{p}^{2} + \frac{1}{2} m \omega ^{2} \beta _{x}^{2} + \alpha _{x} \beta _{x} – \alpha _{p} \beta _{p} – 2 \alpha _{xp} \beta _{x} \beta _{p}, \end{aligned}$$
(13)
defines a new reference frame,
$$\begin{aligned} \vert {\psi _{2}} \rangle = e^{-\frac{i}{\hbar } l t} \vert {\psi _{3}} \rangle , \end{aligned}$$
(14)
with an effective Hamiltonian,
$$\begin{aligned} \hat{H_{3}} = \frac{1}{2m} {\hat{p}}^{2} + \frac{1}{2} m\omega ^{2}{\hat{x}}^{2} + \alpha _{xp} \{{\hat{x}},{\hat{p}}\}, \end{aligned}$$
(15)
that a rotation,
$$\begin{aligned} \vert {\psi _{3}} \rangle = e^{-\frac{i}{\hbar }(\theta _{x}^{2} {\hat{p}}^{2} + \theta _{p}^{2} {\hat{x}}^{2})} \vert {\psi _{4}} \rangle , \end{aligned}$$
(16)
with parameters fulfilling the transcendental equation,
$$\begin{aligned} \tan \left( 4 \theta _{x} \theta _{p} \right) = \frac{m \alpha _{xp}}{\theta _{x}^{2} – m^{2} \omega ^{2} \theta _{p}^{2}} \, 4 \theta _{x} \theta _{p}, \end{aligned}$$
(17)
translates into the standard harmonic oscillator,
$$\begin{aligned} {\hat{H}}_{4} = \frac{1}{2 M} {\hat{p}}^{2} + \frac{1}{2} M \Omega ^{2} {\hat{x}}^{2}, \end{aligned}$$
(18)
with effective mass and frequency,
$$\begin{aligned} \begin{aligned} M =&2m \left[ 1 + m^{2} \omega ^{2} \frac{\theta _{p}^{2}}{\theta _{x}^{2}} + \left( 1 – m^{2} \omega ^{2} \frac{\theta _{p}^{2}}{\theta _{x}^{2}} \right) \cos \left( 4 \theta _{x} \theta _{p} \right) + 4 m \alpha _{xp} \frac{\theta _{p}}{\theta _{x}} \sin \left( 4 \theta _{x} \theta _{p} \right) \right] ^{-1}, \\ \Omega ^{2} =&\frac{1}{2 m M} \frac{\theta _{x}^{2}}{\theta _{p}^{2}}\left[ 1 + m^{2} \omega ^{2} \frac{\theta _{p}^{2}}{\theta _{x}^{2}} – \left( 1 – m^{2} \omega ^{2} \frac{\theta _{p}^{2}}{\theta _{x}^{2}} \right) \cos \left( 4 \theta _{x} \theta _{p} \right) – 4 m \alpha _{xp} \frac{\theta _{p}}{\theta _{x}} \sin \left( 4 \theta _{x} \theta _{p} \right) \right] , \end{aligned} \end{aligned}$$
(19)
in terms of the real parameters from our original general quadratic time-independent QHO model. At this point, it is important to realize that three simple parameter choices are available at hand. First, we may choose to perform the rotation with just the position term, \(\theta _{p} = 0\), to obtain,
$$\begin{aligned} \begin{aligned} \theta _{x}^{2}&= m \alpha _{xp} , \\ \lim _{\theta _{p} \rightarrow 0} M&= m , \\ \lim _{\theta _{p} \rightarrow 0} \Omega ^2&= \omega ^2 – 4 \alpha _{xp}^{2} , \\ \end{aligned} \end{aligned}$$
(20)
a harmonic oscillator with effective mass identical to that in the original frame and effective frequency that tells us the dimensional external scaling should not overcome a fourth of the resonant frequency, \(\omega ^{2} > 4 \alpha _{xp}^2\), in order to have a proper harmonic oscillator in the diagonal frame. We may choose to perform the rotation with just the momentum term, \(\theta _{x} = 0\),
$$\begin{aligned} \begin{aligned} \theta _{p}^{2}&= -\frac{\alpha _{xp}}{m \omega ^{2}} , \\ \lim _{\theta _{x} \rightarrow 0} M&= \frac{m \omega ^2}{\omega ^2 – 4 \alpha _{xp}^{2}} , \\ \lim _{\theta _{x} \rightarrow 0} \Omega ^2&= \omega ^2 – 4 \alpha _{xp}^{2} , \\ \end{aligned} \end{aligned}$$
(21)
and recover an effective mass scaled by the effective frequency square, that is identical to the later case. We may take the parameter values equal in magnitude, \(\theta _{x} = \theta _{p} = \theta\), and realize that the effective frequency remains the same but the effective mass becomes more complicated in shape. We feel it is important to stress that the absolute value of the dimensional external scaling \(\alpha _{xp}\) must be restricted to half the value of the free oscillator frequency, \(\vert \alpha _{xp} \vert < \omega / 2\), in order deal with a harmonic oscillator and avoid dealing with a free particle or an oscillator with complex effective frequency.
Time-dependent model
Now, let us focus on our general quadratic time-dependent QHO Hamiltonian, Eq. (1),
$$\begin{aligned} {\hat{H}}(t) = \frac{1}{2 m(t)} {\hat{p}}^{2} + \frac{1}{2} m(t) \omega ^{2}(t) {\hat{x}}^{2} + \alpha _{x}(t) {\hat{x}} + \alpha _{p}(t) {\hat{p}} + \alpha _{xp}(t) \left\{ {\hat{x}}, {\hat{p}} \right\} + \alpha _{0}(t), \end{aligned}$$
where the units of all the coefficients are the same as in the time-independent Hamiltonian but they are now well-behaved functions of time.
In order to diagonalize our Hamiltonian, we follow a road map similar to the time-independent diagonalization. Our first transformation, again, moves into a rotating frame,
$$\begin{aligned} \vert {\Psi } \rangle = e^{-\frac{i}{\hbar } \int _0^t \alpha _{0} \left( \tau \right) d \tau } \vert {\psi _{1} } \rangle , \end{aligned}$$
(22)
yielding an effective Hamiltonian,
$$\begin{aligned} {\hat{H}}_1(t) = \frac{1}{2 m(t)} {\hat{p}}^{2} + \frac{1}{2} m(t) \omega ^{2}(t) {\hat{x}}^{2} + \alpha _{x} (t) {\hat{x}} + \alpha _{p} (t) {\hat{p}} + \alpha _{xp} (t) \{ {\hat{x}} , {\hat{p}} \}, \end{aligned}$$
(23)
without the time-dependent energy bias term. Now, a time-dependent translation,
$$\begin{aligned} \vert {\psi _{1}} \rangle = e^{- \frac{i}{\hbar } \left[ \beta _{p}(t) {\hat{x}} + \beta _{x}(t) {\hat{p}} \right] } \vert {\psi _2} \rangle , \end{aligned}$$
(24)
with parameters solving the differential equation set,
$$\begin{aligned} \begin{aligned} \ddot{\beta }_{x}(t)&+ \frac{{\dot{m}} (t)}{m (t)} {\dot{\beta }}_{x} (t) – \left[ 2 {\dot{\alpha }}_{xp} (t) – \omega ^{2} (t) + 4 \alpha _{xp}^{2} (t) + \frac{2 \alpha _{xp} {\dot{m}} (t)}{m (t)} \right] \beta _{x} (t) \ldots \\&={\dot{\alpha }}_{p}(t) – \frac{\alpha _{x}(t)}{m(t)} + 2 \alpha _{p}(t) \alpha _{xp}(t) + \alpha _{p}(t) \frac{{\dot{m}}(t)}{m(t)},\\ \beta _{p}(t)&= m (t) \left[ – {\dot{\beta }}_{x}(t) + 2 \alpha _{xp}(t) \beta _{x}(t) + \alpha _{p}(t) \right] , \end{aligned} \end{aligned}$$
(25)
where the boundary conditions are given by Eq. (11) in the time-independent case. It helps us obtain an effective Hamiltonian without the linear driving of the configuration variables,
$$\begin{aligned} {\hat{H}}_{2} (t) = \frac{1}{2 m (t)} {\hat{p}}^{2} + \frac{1}{2} m (t) \omega ^{2} (t) {\hat{x}}^{2} + \alpha _{xp} (t) \{ {\hat{x}}, {\hat{p}} \} + \ell (t), \end{aligned}$$
(26)
where the accumulation time-dependent energy bias term,
$$\begin{aligned} \ell (t) = \frac{1}{2 m (t)} \beta _{p}^{2} (t) + \frac{1}{2} m (t) \omega ^{2} (t) \beta _{x}^{2} (t) + \alpha _x (t) \beta _{x} (t) – \alpha _{p} (t) \beta _{p} (t) – 2 \alpha _{xp} (t) \beta _{x} (t) \beta _{p} (t), \end{aligned}$$
(27)
commutes with all other terms, and help’s define a rotating frame,
$$\begin{aligned} \vert {\psi _{2}} \rangle = e^{ – \frac{ i }{ \hbar } \int _0^{t} \ell ( \tau ) d \tau } \vert {\psi _{3}} \rangle , \end{aligned}$$
(28)
which yields the effective Hamiltonian,
$$\begin{aligned} {\hat{H}}_{3} \left( t \right) = \frac{ 1 }{ 2 m (t) } {\hat{p}}^{2} + \frac{ 1 }{ 2 } m(t) \omega ^{2} (t) {\hat{x}}^{2} + \alpha _{xp} (t) \{ {\hat{x}} , {\hat{p}} \}, \end{aligned}$$
(29)
that is the well-known time-dependent QHO with an extra driving in the anti-commutation term; that is, a driven parametric QHO. Here, we need to take a slight deviation from the previous road map and use the squeezing transformation,
$$\begin{aligned} \vert {\psi _{3}} \rangle = e^{- \frac{i}{\hbar } \frac{\ln \left[ \gamma \left( t \right) \right] }{2} \left\{ {\hat{x}} , {\hat{p}} \right\} } \vert {\psi _{4}}\rangle , \end{aligned}$$
(30)
where the new dimensionless parameter,
$$\begin{aligned} \gamma ^{2} (t) = \frac{m(0)\omega (0)}{m (t) \omega (t)}, \end{aligned}$$
(31)
given in terms of the time-dependent mass and frequency of the system as well as their initial time values, yields the effective parametric QHO Hamiltonian,
$$\begin{aligned} {\hat{H}}_4 (t) = \frac{\omega (t)}{\omega (0)} \left[ \frac{1}{2 m(0) } {\hat{p}}^{2} + \frac{1}{2} m(0) \omega ^{2}(0) {\hat{x}}^2 \right] + \frac{1}{2}\kappa (t) \left\{ {\hat{x}}, {\hat{p}}\right\} , \end{aligned}$$
(32)
where we define an effective parametric drive,
$$\begin{aligned} \kappa (t) = \frac{1}{2}\left[ \frac{{\dot{m}}(t)}{m(t)} + \frac{{\dot{\omega }}(t)}{\omega (t)} + 4\alpha _{xp}(t) \right] , \end{aligned}$$
(33)
for the sake of space. The first right-hand-side of this effective Hamiltonian is the factor of a time-dependent term and a time-independent QHO term suggesting a rotation,
$$\begin{aligned} \vert {\psi _4} \rangle = e^{- \frac{i}{\hbar \omega (0)} \left[ \frac{{\hat{p}}^{2}}{2 m(0)} + \frac{1}{2} m(0) \omega ^{2}(0) {\hat{x}}^2 \right] \frac{\pi }{4}} \vert {\psi _{5}} \rangle , \end{aligned}$$
(34)
yielding the standard time-dependent QHO,
$$\begin{aligned} {\hat{H}}_{5} (t) = \frac{1}{2 m_{5}(t)} {\hat{p}}^{2} + \frac{1}{2} m_{5}(t) \omega _{5}^{2}(t) {\hat{x}}^2, \end{aligned}$$
(35)
with effective mass and frequency,
$$\begin{aligned} \begin{aligned} m_{5}(t)&= \frac{m(0)\omega (0)}{\omega (t) + \kappa (t) }, \\ \omega _{5}^{2}(t)&= \omega ^{2}(t) – \kappa ^2(t) , \end{aligned} \end{aligned}$$
(36)
where we must be careful to work with well-behaved functions that provide us with positive effective parameters, \(m_{5}(t) >0\) and \(\omega _{5}^{2}(t) > 0\). Now, we may factorize the time-dependence using a transformation composed by a squeezing and a rotation66,
$$\begin{aligned} \vert {\psi _{5}} \rangle = e^{\frac{i}{\hbar } \frac{m_{5} (t) {\dot{\rho }} (t)}{2 \rho (t)} {\hat{x}}^{2}} e^{-\frac{i}{\hbar } \frac{\ln \left[ \frac{\rho \left( t \right) }{\rho (0)} \right] }{2}\{{\hat{x}},{\hat{p}}\}} \vert {\psi _{6}} \rangle , \end{aligned}$$
(37)
in that order, where we introduce the auxiliary function \(\rho \left( t\right)\) fulfilling the Ermakov equation,
$$\begin{aligned} {\ddot{\rho }} (t) + \frac{{\dot{m}}_{5} (t)}{m_{5} (t)} {\dot{\rho }} (t) + \omega _{5}^{2} (t) \rho (t) = \frac{1}{m_{5}^{2} (t) \rho ^{3} (t)}. \end{aligned}$$
(38)
with boundary conditions,
$$\begin{aligned} \begin{aligned} \rho (0)&= \frac{1}{\sqrt{m_{5} (0)\omega _{5} (0)}}, \\ {\dot{\rho }}(0)&= 0. \end{aligned} \end{aligned}$$
(39)
that provides us with a diagonal Hamiltonian,
$$\begin{aligned} {\hat{H}}_{6}\left( t\right) = \frac{1}{\omega _{5} (0) m_{5} (t) \rho ^{2} (t)}\left[ \frac{1}{2 m_{5} (0)} {\hat{p}}^{2} + \frac{1}{2} m_{5} (0) \omega _{5}^{2} (0) {\hat{x}}^{2} \right] , \end{aligned}$$
(40)
where the time dependence is factorized from the operators. The second factor in the right-hand-side is a time-independent QHO that yields an effective Hamiltonian,
$$\begin{aligned} {\hat{H}}_{6}(t) = \hbar \Omega (t) \left( {\hat{a}}^{\dagger } {\hat{a}} + \frac{1}{2} \right) , \end{aligned}$$
(41)
in second quantization with effective time-dependent frequency,
$$\begin{aligned} \Omega (t) = \frac{\omega (t) + \kappa (t)}{m(0)\omega (0)\rho ^{2}(t)}, \end{aligned}$$
(42)
with creation and annihilation operators,
$$\begin{aligned} \begin{aligned} {\hat{a}}&= \frac{\rho (0)}{\sqrt{2 \hbar }}\left( \frac{1}{\rho ^2(0)}{\hat{x}} + i{\hat{p}} \right) , \\ {\hat{a}}^\dagger&= \frac{\rho (0)}{\sqrt{2 \hbar }}\left( \frac{1}{\rho ^2(0)}{\hat{x}} – i{\hat{p}} \right) , \end{aligned} \end{aligned}$$
(43)
provided by the initial value of the Ermakov parameter with units of \(\text{mass}^{-1/2} \cdot \text{frequency}^{-1/2}\). We want to stress that we use this second quantization with effective time-independent mass and frequency in the diagonal frame just to recover an expression for the time-dependent
effective
frequency. This mathematical gimmick will not be used for the calculation of expectation values or their variances that will be calculated in first quantization.
At this point, it is straightforward to write the time evolution of the initial state of the system in the original frame,
$$\begin{aligned} \vert {\Psi (t)} \rangle&= {\hat{U}}_{1} (t) {\hat{U}}_{2} (t) {\hat{U}}_{3} (t) {\hat{U}}_{4} (t) {\hat{U}}_{5} (t) {\hat{U}}_{6} (t) {\hat{U}}_{3}^{\dagger } (0) {\hat{U}}_{1}^{\dagger } \left( 0 \right) \vert {\Psi } \left( 0 \right) \rangle , \end{aligned}$$
(44)
in terms of six unitary transformations,
$$\begin{aligned} \begin{aligned} {\hat{U}}_{1} (t)&= e^{- \frac{i}{\hbar } \left[ \beta _{x} (t) {\hat{p}} + \beta _{p} (t) {\hat{x}} \right] },\\ {\hat{U}}_{2} (t)&= e^{-\frac{i}{\hbar } \ln {\left[ \sqrt{\frac{\rho (t)}{\rho (0)} \gamma (t)}\right] } \{{\hat{x}}, {\hat{p}}\}}, \\ {\hat{U}}_{3} (t)&= e^{-\frac{i}{\hbar } \left[ \frac{1}{2 m(0) \omega (0) m_{5} (0) \omega _{5} (0) \rho ^{2} (t)}{\hat{p}}^{2} + \frac{1}{2} m(0) \omega (0) m_{5} (0) \omega _{5} (0) \rho ^{2} (t) {\hat{x}}^{2} \right] \frac{\pi }{4}}, \\ {\hat{U}}_{4} (t)&= e^{\frac{i}{\hbar } \frac{m_{5} (0) \omega _{5} (0)}{2}{\dot{\rho }}(t) \rho (t) m_{5} (t) {\hat{x}}^{2}},\\ {\hat{U}}_{5} (t)&= e^{-\frac{i}{\hbar } \int _0^t \{ \alpha _{0} \left( \tau \right) + \ell \left( \tau \right) \} d \tau }, \\ {\hat{U}}_{6} \left( t \right)&= e^{-\frac{i}{\hbar }\left[ \frac{1}{2m_{5} (0) \omega _{5} (0)}{\hat{p}}^{2} + \frac{1}{2}m_{5} (0) \omega _{5} (0) {\hat{x}}^{2} \right] \int _0^t \Omega (\tau ) d\tau }, \end{aligned} \end{aligned}$$
(45)
where we group together and simplify some of the transformations defined above, that helps us calculate the expectation values for both position and momentum,
$$\begin{aligned} \begin{aligned} \langle {\hat{x}}(t) \rangle =&A(t) \left[ \langle {\hat{x}}(0) \rangle – \beta _{x}(0)\right] + B(t) \left[ \langle {\hat{p}}(0) \rangle + \beta _{p}(0)\right] + \beta _{x} (t), \\ \langle {\hat{p}}(t) \rangle =&D(t) \left[ \langle {\hat{x}}(0) \rangle – \beta _{x}(0)\right] + E(t) \left[ \langle {\hat{p}}(0) \rangle + \beta _{p}(0)\right] – \beta _{p} (t), \end{aligned} \end{aligned}$$
(46)
where we use the notation \(\langle {\hat{o}}(\tau ) \rangle = \langle \psi (\tau ) \vert {\hat{o}} \vert \psi (\tau ) \rangle\) for the expectation values and define the following auxiliary functions,
$$\begin{aligned} \begin{aligned} A(t)&= \frac{\xi (t)}{2} \left\{ \left[ \eta \rho ^{2}(0) + \frac{\varepsilon (t)\rho ^{2}(0)}{\rho ^{2}(t)} – \frac{1}{\eta \rho ^{2}(t)} \right] S(t) + \left[ 1 + \frac{\varepsilon (t)}{\eta \rho ^{2}(t)} + \frac{\rho ^{2}(0)}{\rho ^{2}(t)} \right] C(t) \right\} ,\\ B(t)&= \frac{\xi (t)}{2\eta } \left\{ \left[ \eta \rho ^{2}(0) + \frac{\varepsilon (t)\rho ^{2}(0)}{\rho ^{2}(t)} + \frac{1}{\eta \rho ^{2}(t)} \right] S(t) – \left[ 1 + \frac{\varepsilon (t)}{\eta \rho ^{2}(t)} – \frac{\rho ^{2}(0)}{\rho ^{2}(t)} \right] C(t) \right\} ,\\ D(t)&= \frac{\eta }{2\xi (t)}\left\{ \left[ 1 + \frac{\varepsilon (t)}{\eta \rho ^{2}(0)} – \frac{\rho ^{2}(t)}{\rho ^{2}(0)} \right] C(t) – \left[ \eta \rho ^{2}(t) – \varepsilon (t) + \frac{1}{\eta \rho ^{2}(0)} \right] S(t) \right\} , \\ E(t)&= \frac{1}{2\xi (t)}\left\{ \left[ 1 – \frac{\varepsilon (t)}{\eta \rho ^{2}(0)} + \frac{\rho ^{2}(t)}{\rho ^{2}(0)} \right] C(t) – \left[ \eta \rho ^{2}(t) – \varepsilon (t) – \frac{1}{\eta \rho ^{2}(0)} \right] S(t) \right\} , \\ \end{aligned} \end{aligned}$$
(47)
with parameter set,
$$\begin{aligned} \begin{aligned} \eta&= m(0)\omega (0), \\ \xi (t)&= \frac{\rho (t)}{\rho (0)}\gamma (t), \\ S (t)&= \sin {\left( \int _{0}^t \Omega \left( \tau \right) d\tau \right) }, \\ C (t)&= \cos {\left( \int _{0}^t \Omega \left( \tau \right) d\tau \right) }, \\ \varepsilon (t)&= m_{5}(t){\dot{\rho }}(t)\rho (t), \end{aligned} \end{aligned}$$
(48)
in terms of the functions used to diagonalize the time-dependent Hamiltonian. The variances for position and momentum,
$$\begin{aligned} \begin{aligned} \sigma _{x}^{2}(t) =&A^{2}(t) \sigma _{x}^{2}(0) + B^{2}(t) \sigma _{p}^{2}(0) + A(t)B(t)\left[ \langle \left\{ {\hat{x}}(0), {\hat{p}}(0) \right\} \rangle – 2 \langle {\hat{x}}(0) \rangle \langle {\hat{p}}(0) \rangle \right] , \\ \sigma _{p}^{2}(t) =&D^{2}(t) \sigma _{x}^{2}(0) + E^{2}(t) \sigma _{p}^{2}(0) + D(t)E(t)\left[ \langle \left\{ {\hat{x}}(0), {\hat{p}}(0) \right\} \rangle – 2 \langle {\hat{x}}(0) \rangle \langle {\hat{p}}(0) \rangle \right] . \end{aligned} \end{aligned}$$
(49)
are given in terms of the expected values of position and momentum, \(\langle {\hat{x}}(0) \rangle\) and \(\langle {\hat{p}}(0) \rangle\), and their variances, \(\sigma _{x}^{2}(0)\) and \(\sigma _{p}^{2}(0)\), for the initial state. These expectation values and variances provide a characterization for the time-evolution of initial states where it is straightforward to identify scaling related to the auxiliary function \(\xi (t)\) and rotations to S(t) and C(t).