# Conceptual Problems in Quantum Squeezing

Bogdan Mielnik† Department of Physics
CINVESTAV-IPN
A.P. 14-740
Mexico City 07000, Mexico.  Jesús Fuentes  LCSB
University of Luxembourg
Belvaux L-4367, Luxembourg<EMAIL_ADDRESS>This manuscript was submitted
subsequent to the unfortunate passing of Prof. Mielnik

###### Abstract.

In studies of quantum squeezing, the emphasis is typically placed more on
specific squeezed states and their evolution rather than on the dynamical
operations that could simultaneously squeeze a broader range of quantum
states, regardless of their initial configuration. We explore new developments
in this area, facilitated by gently acting external fields which might induce
squeezing of the canonical observables $q$ and $p$ in charged particles. The
extensive research in this field has yielded many valuable insights, raising
the question of whether there is still room for significant new contributions
to our understanding. Nonetheless, we present some exactly solvable instances
of this problem, observed in symmetric evolution intervals. These intervals
allow for the explicit determination of the temporal dependence of external
fields necessary to generate the required evolution operators. Our findings
are linked with a straightforward application of Toeplitz matrices, offering a
more accessible description of the problem compared to the frequently employed
Ermakov-Milne invariants.

###### Key words and phrases:

Quantum dynamics, quantum squeezing, quantum operations

###### 1991 Mathematics Subject Classification:

Primary 8100, 81S07, 81Q93; Secondary 81R15

##  1\. Introduction

In traditional control problems in quantum theory, two principal frameworks
are often juxtaposed: Schrödinger’s picture, focusing on state evolution, and
Heisenberg’s, emphasising the evolution of observables. Dirac robustly
advocated for the merits of Heisenberg’s representation in a notable polemic
article [14]. Regardless of one’s stance on his critique, certain practical
aspects of his arguments find relevance in contemporary quantum control
challenges. This is particularly true for linear transformations of canonical
variables $q$ and $p$ including squeezing, as described by Yuen [54]. Yuen’s
application demonstrates the potential for precise and rapid measurement of
free particle position using twisted canonical observables [55]. His
techniques in squeezing presaged the advent of quantum tomography [33, 34, 3],
which represents quantum states through tomographic images derived from the
Radon transforms of canonical variables [44, 17]. This field is currently
experiencing a surge of innovative ideas and contributions [31]. Some of the
consequences of this standpoint for basic quantum mechanical ideas, though not
immediate, are not negligible.

Quantum mechanics formalism posits that every self-adjoint operator in Hilbert
space corresponds to a measuring apparatus, capable of an instantaneous
assessment, collapsing the wave packet into one of its eigenstates. However,
practical laboratory experience suggests a more challenging scenario where
measurement is preceded by an auxiliary evolution, and the fundamental
indeterministic decision occurs at the final moment.

Our focus is particularly on auxiliary operations such as amplification or
squeezing, exploring the prospects of applying Thorne et al.’s concept of
demolition-free measurements [48, 49]. It is important to note, however, that
not all available techniques inspire unequivocal confidence. This includes
issues related to decoherence [7], instability [26], and delayed choice [51].
Recently, even entangled states and their radiation effects have come under
scrutiny [30, 32, 16, 43].

For these reasons, we focus exclusively on soft control techniques that avoid
abrupt changes and minimise radiative pollution. Although our problem is
elementary, it is not entirely trivial. In existing theory, it is not always
assured that every well-defined unitary operator can be dynamically realised,
or at least approximated, by realistic motion generators. Hence, we propose a
simple class of time-dependent Hamiltonians to investigate whether they can
effectively generate true squeezing effects in massive particles. We regard
variable external fields as the sole plausible source of this phenomenon.
Consequently, we disregard formal results derived for time-dependent masses
and material constants, among others. Nevertheless, the array of impressive
existing results cannot be overlooked [15]. We also consciously steer clear of
the potential complexities of quantum field theory [51, 30]. Thus, we aim to
confine our approach to a purely quantum mechanical framework, focusing on
slow (adiabatic) processes that minimise radiative pollution. For analytical
convenience, our calculations were performed using dimensionless variables.

##  2\. Elementary Evolution Matrices

Let us consider a pair of conjugate variables $q$ and $p$, representing the
dimensionless canonical position and momentum, respectively. In this
dimensionless framework, we define $\tau$ as a dimensionless time parameter,
adopting units where the mass $m=1$ and $\hbar=1$, thus ensuring $[q,p]=i$.
Consider an elastic force represented by a nonsingular, bounded $\beta(\tau)$.
The evolution equations generated by the Hamiltonian
$H(\tau)=p^{2}/2+\beta(\tau)q^{2}/2$ imply exactly the same linear equations
for either classical or quantum canonical variables. These are:
$dq/d\tau=p(\tau)$ and $dp/d\tau=-\beta(\tau)q(\tau)$. Over any time interval
$[\tau_{0},\tau]$, this leads to an identical transformation of the classical
or quantum canonical pair. Such transformations are expressed by the same
family of $2\times 2$ symplectic evolution matrices $u(\tau,\tau_{0})$:

$\begin{pmatrix}q(\tau)\\\
p(\tau)\end{pmatrix}=u(\tau,\tau_{0})\begin{pmatrix}q(\tau_{0})\\\
p(\tau_{0})\end{pmatrix};\quad u(\tau_{0},\tau_{0})=1,$ (1)

determined by the matrix equations

$\frac{\text{d}}{\text{d}\tau}u(\tau,\tau_{0})=\Lambda(\tau)u(\tau,\tau_{0});\quad\Lambda(\tau)=\begin{pmatrix}0&&1\\\
-\beta(\tau)&&0\end{pmatrix}.$ (2)

The reciprocity between the classical and quantum pictures does not end up
here. It turns out that, in the absence of spin, each unitary evolution
operator $U(\tau,\tau_{0})$ in $L^{2}(\mathbb{R})$ generated by the
Hamiltonian $H(\tau)$ is determined, up to a phase factor, by the canonical
transformation that it induces. This is the consequence of the following
simple lemma [45, 35, 38, 36]:

######  Lemma 1.

The family of the unitary operators $U(\tau,\tau_{0})$ describing the
evolution generated by the quadratic Hamiltonians $H(\tau)$, is determined, up
to a constant phase factor, by the corresponding matrices $u(\tau,\tau_{0})$.
Consequently, it is also determined by the corresponding classical
trajectories.

###### Proof.

It is sufficient to observe that if two unitary operators $U_{1}$ and $U_{2}$
induce the same transformation of the canonical variables, namely
$U_{1}^{\dagger}qU_{1}=U_{2}^{\dagger}qU_{2}$ and
$U_{1}^{\dagger}pU_{1}=U_{2}^{\dagger}pU_{2}$, then $U_{1}U_{2}^{\dagger}$
commutes with both $q$ and $p$. Consequently, it also commutes with any
function of $q$ and $p$. Given that in $L^{2}(\mathbb{R})$ the functions of
$q$ and $p$ generate an irreducible algebra, $U_{1}U_{2}^{\dagger}$ must be a
$c$-number. Being unitary, it can only be a phase factor,
$U_{1}U_{2}^{\dagger}=e^{i\varphi}\Rightarrow U_{1}=e^{i\varphi}U_{2}$, where
$\varphi\in\mathbb{R}$. ∎

Any two unitary operators that differ only by a $c$-number phase, even though
they act differently on the state vectors, induce the same transformation of
quantum states. Therefore, we shall consider them equivalent, $U_{1}\equiv
U_{2}$. It immediately follows that the trajectories of the classical motion
problem with quadratic $H(\tau)$ fully determine the evolution of pure or
mixed quantum states $\rho=\rho^{\dagger}\geq 0$, $\operatorname{Tr}\rho=1$,
and, modulo equivalence, the entire unitary history, which we denote for
simplicity as $U(\tau)=U(\tau,\tau_{0})$. Our description complements the
trends of phase geometry [5, 2, 20, 9, 52, 10, 6, 12, 53]; while it does not
describe geometric phases, it determines alternative aspects of quantum states
like the motion of centres, packet shapes, and all nuances of the statistical
interpretation.

##  3\. Classification of the Motion of Massive Particles

In quantum optics of coherent photon states, the notion of parametric
amplification, as discussed by Mollow and Glauber [40], plays an important
role. Yet, in the description of massive particles the Heisenberg evolution of
the canonical observables—the trajectory picture—receives less attention, even
though it allows for the extension of optical concepts [52, 53]. This is
particularly interesting for charged particles in ion traps driven by time-
dependent fields, whether or not they coincide with Paul’s formula [42]. The
most interesting here is the case of quite arbitrary periodic potentials.

For a given periodic field $\beta$, where
$\beta({\footnotesize\text{T}}+\tau)=\beta(\tau)$, the most important matrices
(1) are $u({\footnotesize\text{T}}+\tau_{0},\tau_{0})$ describing the repeated
evolution events. Since they are symplectic, their algebraic structure is
uniquely defined by a single number
$\Gamma=\operatorname{Tr}u({\footnotesize\text{T}}+\tau_{0},\tau_{0})$. (We
are not referring to Ermakov-Milne invariants [18, 39, 41].) Though the
matrices $u({\footnotesize\text{T}}+\tau_{0},\tau_{0})$ depend on $\tau_{0}$,
$\Gamma$ does not, permitting to classify the evolution processes generated by
$\beta$ in any periodicity interval. The distinction between the three types
of behaviour is quite elementary:

  1. I

If $|\Gamma|<2$, regardless of the details, the repeated $\beta$-periods
produce an evolution matrix with a pair of eigenvalues $e^{i\sigma}$ and
$e^{-i\sigma}$ ($\sigma\in\mathbb{R},0<\sigma<\frac{\pi}{2}$) generating a
stable (oscillating) evolution process. It allows the construction of the
global creation and annihilation operators $a^{+},a^{-}$ defined by the row
eigenvectors of $u({\footnotesize\text{T}}+\tau_{0},\tau_{0})$, thus
characterising the evolution in the whole periodicity interval [38, 37].

  2. II

If $|\Gamma|=2$, the process generated by $\beta$ falls on the stability
threshold with eigenvalues $\pm 1$ permitting to approximate a family of
interesting dynamical operations, see also [36].

  3. III

If $|\Gamma|>2$ then each single-periodic evolution matrix now has a pair of
real, non-vanishing eigenvalues, $\lambda^{+}=1/\lambda^{-}$ with
$\lambda^{+}=e^{\sigma}$ and $\lambda^{-}=e^{-\sigma}$ producing the squeezing
of the corresponding pair of canonical observables $a^{\pm}$ defined again by
the eigenvectors of $u({\footnotesize\text{T}}+\tau_{0},\tau_{0})$, that is,
$a^{+}$ expands at the cost of contracting $a^{-}$ or vice-versa.

The above global data seem more relevant than the description in terms of the
instantaneous creation and annihilation operators which do not make obvious
the stability thresholds. Specifically, for Paul potentials described by
$\beta(\tau)=\beta_{0}+2\beta_{1}\cos\tau$, the map of the squeezing boundary
is determined by the Strutt diagram [4], traditionally limited to describe the
ion trapping (in stability areas). Out of them are precisely the squeezing
effects in III. To illustrate all this, integrating (2) for the specific case
of Paul’s potential is insightful for $(\beta_{0},\beta_{1})$ out of the
stability domain.

##  4\. Mathieu Squeezing

Figure 1. In a Paul trap, squeezing is represented by the red curve, which
plots the evolution matrices (1) where $u_{12}\to 0$, within the squeezing
region (coloured areas) on the Strutt map. Similarly, the blue curves denote
the pairs $(\beta_{0},\beta_{1})$ for which $u_{21}\to 0$. Points located in
the negative portions of the squeezing trajectories signify the occurrence of
inverted squeezing effects.

As established in previous studies [52, 53], squeezing is unattainable when
$\beta(\tau)$ exhibits symmetry within the operational interval. To
investigate this further, we opted for numerical integration of Eq. (2). For
this purpose, we employed a Paul potential, defined as
$\beta=\beta_{0}+2\beta_{1}\cos\tau$, over the interval
$\left[\pi/2,5\pi/2\right]$. We varied the parameters $\beta_{0}$ and
$\beta_{1}$ in the squeezing region of the Strutt diagram, as reported in
previous works [4, 37].

Following this, we conducted a scanning procedure across regions I-III, with
the aim of identifying those evolution matrices $u$ that produce quantum
squeezing effects. Our findings, illustrated in Figure 1, generalise the
numerical data presented in [37]. In the figure, the clear zones correspond to
the stability domain (class I), while the coloured zones, shaded in yellow,
correspond to the domain of unstable motion (class III), where squeezing
effects occur. The boundaries between these clear and coloured areas furnish
the class of motion II. The continuous red curves represent the combinations
of $\beta_{0}$ and $\beta_{1}$ for which the matrix element $u_{12}=0$ in the
evolution matrix $u=u\left(5\pi/2,\pi/2\right)$. Conversely, the curves in
blue outline the parameter sets $(\beta_{0},\beta_{1})$ such that $u_{21}=0$.
The intersection of both curves, denoted by $u_{s}$, yields the pair
$(\beta_{0},\beta_{1})$ that grants the genuine position and momentum
squeezing transformations, $q\rightarrow\lambda q$ and $p\rightarrow
p/\lambda$, in that order.

The matrix elements $u_{12}$, represented by the values along the red line in
Figure 1, indicate subtle variations in the effects observed when $q$ is
squeezed (or amplified). These variations specifically affect canonical
variables, exemplified by $a^{-}=u_{21}q+1/\lambda p$. To illustrate this, we
selected four distinct evolution matrices. These matrices illustrate varying
cases of squeezing, achieved through four different pairs of
$(\beta_{0},\beta_{1})$, as follows:

$\displaystyle u_{1}$ $\displaystyle=\begin{pmatrix}0.362&\sim 0\\\
-1.114&2.751\end{pmatrix},$ $\displaystyle u_{2}$
$\displaystyle=\begin{pmatrix}0.175&\sim 0\\\ 3.501&5.798\end{pmatrix},$ (3)
$\displaystyle u_{3}$ $\displaystyle=\begin{pmatrix}0.216&\sim 0\\\
5.444&4.833\end{pmatrix},$ $\displaystyle u_{s}$
$\displaystyle=\begin{pmatrix}0.227&\sim 0\\\ \sim 0&4.394\end{pmatrix}.$

Correspondingly, the specific matrices $u_{1},u_{2},u_{3}$, and $u_{s}$ were
derived using the following sets of amplitude pairs:

$(\beta_{0},\beta_{1})={(\pi/3,\pi/5),\,(\pi/2,4\pi/13),\,(9\pi/16,5\pi/11),\,(4\pi/13,11\pi/41)}.$

While the matrix $u_{s}$ at the intersection of both curves in the upper
(positive) part of the diagram represents the coordinate squeezing
$q\rightarrow\lambda q$, $p\rightarrow(1/\lambda)p$ with $\lambda\approx
0.227$, the corresponding intersection on the lower (negative) part represents
an inverse operation with $\lambda^{-1}\approx 4.394$; thus, amplifying $q$
and squeezing $p$. Henceforth, if the corresponding pulses were to be
successively applied to two pairs of electrodes in a cylindric Paul trap, then
the particle state would suffer a sequential expansion of its $x$ variable
with the simultaneous squeezing of $y$, followed by the inverse amplification
of $y$ and squeezing of $x$. It remains an open question whether new squeezing
techniques could emerge from the generalisation of the operational techniques
of high frequency pulses described in [28].

The reader might find the detailed focus on these minor aspects somewhat
tedious. The boring problem of physical units brings, however, additional
information. For the dimensionless time $\tau=\omega t$, consider the
parameter T. In the expression
$\beta(\tau)=e\Phi(t){\footnotesize\text{T}}^{2}/r_{0}^{2}m$, T represents the
period of the oscillating Paul voltage on the trap wall, defined as
$\Phi(t)=\Phi_{0}+\Phi_{1}\cos\omega t$. This leads to
$\beta(\tau)=\beta_{0}+2\beta_{1}\cos\tau$. Consequently, the same
dimensionless matrix $u_{s}$ in (3) is generated in
$\left[\pi/2,5\pi/2\right]$ by physical parameters such that:

$\frac{e\Phi_{0}}{\omega^{2}r_{0}^{2}m}=\beta_{0},\quad\frac{e\Phi_{1}}{\omega^{2}r_{0}^{2}m}=2\beta_{1}.$
(4)

For particles of a fixed mass and charge, the varying quantities are the
potentials $\Phi$ and the physical time ${\footnotesize\text{T}}=2\pi/\omega$
of the operations corresponding to the dimensionless interval
$\left[\pi/2,5\pi/2\right]$. Hence, for any fixed $r_{0}$, the smaller
$\omega$ is, the lower voltages $\Phi_{0}$ and $\Phi_{1}$ required to assure
the same result, provided too weak fields do not permit the particle to escape
or to collide with the trap surfaces. Consider a proton
($m=m_{p}\simeq\text{1.67}\times 10^{-24}$g) in an unusually large ion trap
with $r_{0}=10$cm. If subjected to a moderately oscillating Paul field
corresponding to a 3km long radio wave, one would find
$\omega^{2}r_{0}^{2}m_{p}\simeq
10^{-12}\text{g}\frac{\text{cm}^{2}}{s^{2}}=\text{1.67}\times
10^{-12}\text{g}\frac{\text{cm}^{2}}{\text{s}^{2}}\simeq\text{1.04233}$eV,
this leads to estimated voltages of:
$\Phi_{0}\simeq\text{1.0423}\beta_{0}\text{V}\simeq\text{1.268}$V and
$\Phi_{1}\simeq\text{2.0846}\beta_{1}\text{V}\simeq\text{1.759}$V. In an even
larger trap, with $r_{0}=100$cm or, alternatively, for $r_{0}=10$cm but the
frequency 10 times higher, the voltages needed on the walls should be already
100 times higher!

While the analytic expressions in [21] could yield more exact results, our
computational experiments indeed indicate that the phenomena of $qp$-squeezing
can happen in Paul traps. However, these concern only the extremely clean
oscillations, without any laser cooling, which are crucial for experimental
trapping techniques [42], nor any dissipative perturbations. Moreover, the
squeezing effects described by matrices (3) are transient, manifesting only at
precisely defined moments which makes difficult the observation of the
phenomenon in the oscillating trap fields.

##  5\. The Option of Squeezed Fourier

Mathematically, one of the simplest ways to construct quantum operations is to
apply sequences of external $\delta$-pulses that interrupt a continuous
evolution process—such as the free evolution, the harmonic oscillation, or
further interactions [1, 19]. However, this method is limited by the practical
challenge of applying $\delta$-pulses to external fields. In the case of
squeezing, a more systematic method could involve composing evolution
incidents which belong to the equilibrium zone I but their products do not.
One possibility is to utilise segments of time-independent oscillator fields
with elastic forces $\beta=\kappa^{2}=\text{const.}$, generating symplectic
rotations:

$u=\begin{pmatrix}\cos\kappa\tau&\frac{\sin\kappa\tau}{\kappa}\\\
-\kappa\sin\kappa\tau&\cos\kappa\tau\end{pmatrix}.$ (5)

The simplest instances, obtained when $\cos\kappa\tau=0$, correspond to the
squeezed Fourier transformations

$u=\begin{pmatrix}0&\pm\frac{1}{\kappa}\\\ \mp\kappa&0\end{pmatrix}.$ (6)

Following the proposals in [27] and [24] applying two such steps with
different $\kappa$-values generate the evolution matrix:

$u_{\lambda}=\begin{pmatrix}0&\pm\frac{1}{\kappa_{1}}\\\
\mp\kappa_{1}&0\end{pmatrix}\begin{pmatrix}0&\pm\frac{1}{\kappa_{2}}\\\
\mp\kappa_{2}&0\end{pmatrix}=\begin{pmatrix}\lambda&0\\\
0&\frac{1}{\lambda}\end{pmatrix};\quad\lambda=-\frac{\kappa_{2}}{\kappa_{1}}$
(7)

resulting in the squeezing of the canonical pair: $q\rightarrow\lambda q$,
$p\rightarrow p/\lambda$, with the effective evolution operator:
$U_{\lambda}=\exp[-i\sigma(pq+qp)/2];\quad\sigma=\ln\lambda$. However, this
requires two different values of $\kappa_{1}\neq\kappa_{2}$ within two
distinct time intervals separated by an abrupt potential jump. Here, the times
$\tau_{1}$ and $\tau_{2}$ can fulfil
$\kappa_{1}\tau_{1}=\kappa_{2}\tau_{2}=\pi/2$ to assure that both $\kappa_{1}$
and $\kappa_{2}$ grant two distinct squeezed Fourier operations in their time
intervals. If one wants to apply two potential steps on the null background,
it means at least three jumps
($0\rightarrow\kappa_{1}\rightarrow\kappa_{2}\rightarrow 0$). The precise
method of approximating a jump in the elastic potential remains a question.
Moreover, each $\kappa$-jump indicates an energy transfer to the micro-
particle [24]. Hence, could the pair of generalised Fourier operations in (7)
be superposed in a soft way with an identical final result? In fact, as we
will demonstrate in the following section, recent advances in the inverse
evolution problem reveal the existence of such effects.

##  6\. Toeplitz Matrices and Exact Operations

While the exact expressions in (7) were already known, it was previously
overlooked that they can be produced using the simple anti-commuting algebra
of $2\times 2$ equidiagonal, symplectic matrices $u$, where
$u_{11}=u_{22}=\frac{1}{2}\operatorname{Tr}u$. Remarkably, for any two such
matrices $u$ and $v$, their anti-commutator $uv+vu$, as well as the symmetric
products $uvu$ and $vuv$, are part of the same family.

The Toeplitz matrices, which have spurred considerable research [8, 50, 13],
have not been widely recognised for their fundamental quantum control
applications. In our context, (7) does not require the elimination of jumps.
This allows for greater versatility in constructing squeezed Fourier
operations. These operations are achieved through symmetric products of
multiple small symplectic contributions (5), each operating over different
time intervals with varying $\beta$ fields. Employing segments of symplectic
rotations $v_{k}$, induced by Hamiltonians $H(\tau)$ with specific
$\beta=\beta_{k}$ during time intervals $\Delta\tau_{k}~{}(k=0,1,2,\ldots)$,
allows us to define the symmetric product:

$u=v_{n}\ldots v_{1}v_{0}v_{1}\ldots v_{n}$ (8)

which remains symplectic and equidiagonal—akin to the simplest Toeplitz
class—with matrix elements $u_{11}=u_{22}=\frac{1}{2}\operatorname{Tr}u$. When
(8) achieves $\operatorname{Tr}u=0$, the matrix $u$ transforms into a squeezed
Fourier matrix. Their continuous counterparts are similarly obtainable.

To achieve this, one must assume that the amplitude $\beta(\tau)$ is symmetric
around a certain point $\tau=0$, thereby ensuring $\beta(\tau)=\beta(-\tau)$.
Consequently, by considering the limits of small jumps $du$ caused by applying
contributions $dv=\Lambda(\tau)d\tau$ from both sides, one derives the
differential equation for $u=u(\tau,-\tau)$ within the expanding interval
$[-\tau,\tau]$:

$\frac{du}{d\tau}=\Lambda(\tau)u+u\Lambda(\tau).$ (9)

This anti-commuting form leads easily to an exact solution. Since
$\Lambda(\tau)$ is given by (2), equation (9) becomes [36, 37]:

$\frac{du}{d\tau}=\begin{pmatrix}u_{21}-\beta u_{12}&\operatorname{Tr}u\\\
-\beta\operatorname{Tr}u&u_{21}-\beta u_{12}\end{pmatrix}=(u_{21}-\beta
u_{12})\mathbb{I}+\operatorname{Tr}u\begin{pmatrix}0&1\\\
-\beta&0\end{pmatrix}.$ (10)

For a given symmetric $\beta(\tau)$, the former equation explicitly determines
the matrices $u=u(\tau,-\tau)$ for the expanded time interval $[-\tau,\tau]$
in terms of just one function $\theta(\tau)=u_{12}(\tau,-\tau)$. Indeed, as
(10) suggests the same differential equation for both $u_{11}$ and $u_{22}$,
namely $du_{11}/d\tau=du_{22}/d\tau=u_{21}-\beta u_{12}$, and given that
$u_{11}=u_{22}=1$ at $\tau=0$, it follows that
$u_{11}=u_{22}=\frac{1}{2}\operatorname{Tr}u=\frac{1}{2}\theta^{\prime}(\tau)$
for $u$ in any time interval $[-\tau,\tau]$. Furthermore, as $u=u(\tau,-\tau)$
matrices are symplectic, which means
$\text{Det~{}}u=\left[\theta^{\prime}(\tau)/2\right]^{2}-\theta u_{21}=1$, one
obtains

$u_{21}=\frac{\left[\frac{1}{2}\theta^{\prime}(\tau)\right]^{2}-1}{\theta}.$
(11)

Consequently, (10) specifies the amplitude $\beta(\tau)$ required to generate
the matrices $u=u(\tau,-\tau)$. Besides:

$\beta u_{12}=u_{21}-\frac{du_{11}}{d\tau}.$ (12)

Given that $u_{12}=\theta$, and with $du_{11}/d\tau=\theta^{\prime\prime}/2$
along with $u_{21}$ as specified in (11), we can deduce that:

$\beta=-\frac{\theta^{\prime\prime}}{2\theta}+\frac{\left[\frac{1}{2}\theta^{\prime}(\tau)\right]^{2}-1}{\theta^{2}}.$
(13)

This solves the symmetric evolution problem for $u$ and $\beta$ in any
interval $[-\tau,\tau]$ in terms of one, almost arbitrary function
$\theta(\tau)$, restricted by non-trivial conditions at single points only.
Hence, (13) is indeed an exact solution of the inverse evolution problem,
offering $\beta(\tau)$ in terms of the function
$\theta(\tau)=u_{1,2}(\tau,-\tau)$ representing the evolution matrices for the
expanding (or shrinking) evolution intervals $[\tau,-\tau]$. Note though that
the dependence of $u(\tau,\tau_{0})$ on $\beta(\tau)$ given by (13) in any
non-symmetric interval $[\tau,\tau_{0}]$ requires still an additional
integration of (2) between $\tau_{0}$ and $\tau$. Some simple algebraic
relations of $\beta$ and $\theta$ are worth attention.

######  Lemma 2.

Let $\beta(\tau)$ be defined as in (13), over an interval $[-T,T]$, where
$T\in\mathbb{R}$. Further assume that $\theta(\tau)$ is a continuous function
and, at least, three times differentiable over the same interval. Then, the
following conditions ensure the continuity and differentiability of $\beta$,
as well as the dynamical relations between $\theta$ and $\beta$:

  1. i.

At any point $\tau$ where $\theta(\tau)=0$, it must be the case that
$\theta^{\prime}(\tau)=\pm 2$.

  2. ii.

If $\theta^{\prime\prime\prime}(\tau)=0$, then $\beta^{\prime}(\tau)=0$.

  3. iii.

At any point $\tau$ where $\theta(\tau)\neq 0$ but $\theta^{\prime}(\tau)=0$,
Eq. (10) for the interval $[-\tau,\tau]$ represents the squeezed Fourier
transformation with $\beta(\tau)$ at the endpoints satisfying
$\beta(\tau)\theta^{2}=-\frac{1}{2}\theta^{\prime\prime}\theta-1$.

###### Proof.

The proof follows directly from (12). Specifically, (10) and the initial
condition ensure that $u_{11}=u_{22}=\frac{1}{2}\theta^{\prime}(\tau)$.
Therefore, when $\theta^{\prime}=0$, both $u_{11}=u_{22}=0$, leading to
$u_{12}=b\neq 0$ and $u_{21}=-1/b$, which corresponds to the general form of
the squeezed Fourier transformation outlined in Section 5. Concurrently, (13)
simplifies to yield

$\beta(\tau)\theta^{2}=-\frac{1}{2}\theta^{\prime\prime}\theta-1\implies\beta(\tau)b+\frac{\theta^{\prime\prime}}{2}+\frac{1}{b}=0.$

In particular, if $\theta^{\prime\prime}(\tau)b=-2$, then $\beta(\tau)=0$.
This completes the proof. ∎

A certain curious quid pro quo should be noted. Without entering into the
phase problems [11, 12, 25], we focused on the simplest case of Toeplitz
matrices [8, 50]. This approach, also employed in [38, 36, 37], resolves the
inverse evolution problem for $\beta(\tau)=\kappa^{2}(\tau)$ in terms of
$\theta(\tau)$, without relying on any auxiliary invariants. However, its
purely comparative sense, should be stressed. For a fixed pair of canonical
variables $q,p$ it does not give the causally progressing process of the
evolution, but rather compares the evolution incidents in a family of
expanding intervals $[-\tau,\tau]$. Should one wish to follow the causal
development of the classical or quantum systems, the Ermakov-Milne equation
[18, 39] might be useful. An interrelation between both methods waits still
for an exact description. It is not excluded that anti-commutator algebras
could also be helpful in some higher-dimensional canonical problems.

##  7\. Imperfections and Open Problems

#### Troubles With Geometry

In numerous laboratories, techniques employed for maintaining and cooling ions
are adequate for studying atomic structures, but they fall short in broader
applications. The time-dependent oscillator potential is often created on a
highly local scale—specifically, in the immediate vicinity of the central axis
of a quadrupole trap, which may consist of merely four metal bars [46]. The
opportunity to extend the reach of oscillator fields for controlling unitary
evolution arises either in conventional or cylindrical Paul traps with ideally
hyperbolic surfaces, or within the interiors of cylindrical solenoids,
provided that the operational area is sufficiently large. If this condition is
met, the propagation of the controlling field within the trap introduces an
additional challenge.

#### Control of Currents

While the latter addresses the spatial constraints, it leaves unanswered the
question of inducing temporally variable yet spatially homogeneous surface
currents independent of the axis coordinate $z$. In the static case, the
magnetic field $\mathbf{B}$ within a solenoid is generated by a stationary
current $\mathbf{J}$ circulating around the solenoid’s surface. Integrating
the magnetic field along a closed contour around the solenoid leads to the
well-known expression $B=\frac{4\pi}{c}\frac{\Delta J}{\Delta z}$. However,
generating time-dependent currents that are uniform across all surface
sections, irrespective of $z$, remains a challenge. If a solenoid were
constructed from a single spiral wire wrapped around its cylindrical surface
and connected at both ends to a potential difference $\Phi(t)$, any minor
alteration in $\Phi$ would propagate as a current pulse along the solenoid,
thereby creating $z$-dependent fields instead of a quasi-static $B(t)$. A
potential workaround might involve covering the cylindrical surface with
shorter wires connected to a common variable voltage source, though this is
not the only possible solution.

#### The Model of a Rotating Cylinder

An alternative approach is inspired by an example described by Griffiths [23].
Consider a cylindrical surface made of a non-conducting material (such as
glass) with a radius $R$, uniformly charged with a surface density $\sigma$.
Each circular belt of 1cm height would thus carry a charge proportional to
$R\sigma$. Such an experimental setup is feasible. If the cylinder has a
radius $R=20$cm and rotates at a frequency $\omega=1\,\text{s}^{-1}$ around
its axis $z$, with each 1cm horizontal belt carrying a charge of 1C, it would
generate a homogeneous magnetic field $\mathbf{n}B$ with an intensity given by

$B=\frac{4\pi}{c}\omega R\sigma\approx 1.25\,\text{G},$ (14)

at least under the post-post-Newtonian approximation. By varying the angular
velocity $\omega=\omega(t)$, one could create a practically homogeneous
magnetic field $\mathbf{n}B(t)$ within a quasi-static environment. The
efficacy of such techniques remains an open question.

#### Time Control

To induce a quantum state operation via variable fields, the micro-object must
be exposed to these fields for a precisely defined interval, from the
operation’s commencement to its conclusion. This is especially critical for
operations driven by time-dependent magnetic fields. The charged particle must
enter the solenoid in a known initial state at the exact moment a Fourier
squeezing operation begins and must be assessed at a specific instant after
one or multiple field pattern applications. This synchronisation requirement
is often overlooked in quantum control literature. One might consider a long,
yet finite, solenoid. Inject a particle at an exact moment into the solenoid
at a point $z=z_{0}$ with a velocity $v_{z}$. The particle’s wave packet
propagates and alters its form through the solenoid, eventually arriving at
the other end during the time span of one or multiple squeezing operations. A
measuring device, such as a photographic plate, records its position on a new
orthogonal plane. Several errors are inevitable, arising from the granularity
of the detection system and the Heisenberg uncertainty principle, which
affects the time of flight. An intriguing feature is that if the squeezing
operation amplifies coordinates while shrinking momenta, this is akin to a
non-demolition measurement [48, 49], albeit with increased precision in
deducing the initial coordinates.

#### Neglected Perturbations

Our calculations focus solely on pure particle states evolving in slowly
varying external fields, while neglecting potential interactions with any
residual matter in ion traps or solenoids. We have also omitted the direct
packet reflection or absorption by laboratory walls, thus leaving questions
open regarding the necessary dimensions of the ion trap or solenoid, as well
as the unresolved issue of a time operator, even in the context of flat
surfaces. Other potential modifications by dissipative mechanisms such as
those described by Lindblad, Gisin, and Percival have also been ignored [29,
22, 47]. Several unresolved and perhaps contentious aspects in quantum
mechanics remain. Despite these, one should not forget that our techniques are
grounded in orthodox quantum mechanics, specifically the evolution matrices
(1). These low-energy phenomena may be as fundamental to the field of quantum
theory as their high-energy counterparts.

### Acknowledgments

We acknowledge the discussions with Dr. Jan Gutt from Physics Department of
Polish Academy of Science. J.F. wishes to express profound gratitude to the
late Prof. Mielnik, whose intellectual contributions during the period of
2015-2018 were instrumental in shaping the foundations of this work. It is
with a sense of deep respect and acknowledgment that this manuscript is
submitted after his decease.

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