# Behavior of correlation functions in the dynamics of the Multiparticle
Quantum Arnol’d Cat

Giorgio Mantica
Center for Non-linear and Complex Systems
Università dell’Insubria, Via Valleggio 11, Como, ITALY
and INDAM, GNFM,
and I.N.F.N. sezione di Milano

###### Abstract

The multi-particle Arnol’d cat is a generalization of the Hamiltonian system,
both classical and quantum, whose period evolution operator is the renown map
that bears its name. It is obtained following the Joos-Zeh prescription for
decoherence, by adding a number of scattering particles in the configuration
space of the cat. Quantization follows swiftly, if the Hamiltonian approach,
rather than the semiclassical, is adopted. I have studied this system in a
series of previous works, focusing on the problem of quantum-classical
correspondence. In this paper I test the dynamics of this system by two
related yet different indicators: the time autocorrelation function of the
canonical position and the out of time correlator of position and momentum.

##  1 Introduction: the Classical Limit of Quantum Mechanics, the
Correspondence Principle and Decoherence

Since the very beginning of quantum mechanics the problem has been posed about
the so-called classical limit, if for nothing else to give a meaning to the
measurement process [1]. In parallel the correspondence principle was
introduced. This principle comprises two different concepts, yet closely
intertwined. They appear clearly stated by Dirac in [2]: says he that
classical dynamics is employed in Bohr’s theory via [..] the assumption,
called the Correspondence Principle, that the classical theory gives the right
results in the limiting case when the action per cycle of the system is large
compared to Planck’s constant h. And, a few lines after: In a recent paper
Heisenberg puts forward a new theory which suggests that it is not the
equations of classical mechanics that are in any way at fault, but that the
mathematical operations by which physical results are deduced from them that
require modification. Therefore, correspondence can be interpreted in two
ways: according to the latter sentence, classical canonical variables can be
turned into quantum mechanical operators and classical Hamilton equation into
Heisenberg’s; according to the former, once this correspondence has been
established, quantum dynamics should yield classical results for large values
of the action. Dirac concludes the quoted paragraph by adding: All the
information supplied by the classical theory can thus be made use of in the
new theory (emphasis in the original).

In a series of papers with Ford in the last century we raised the doubt that
not all the dynamical content of classical dynamics can be retrieved by
quantum dynamics in the classical limit [3, 4]. In parallel with this
theoretical question is the more practical attitude of considering the
specific details of such correspondence: in other words, to find in which
regimes and with what accuracy agreement of the two results is to be expected,
finally, to ask whether this formal correspondence (formal, because it can be
stated at a fully abstract level) is capable of describing physical phenomena
well predicted by classical dynamics.

This situation can also be described as follows: While the correspondence
principle affirms that the mathematical operations of quantum mechanics yield
results that, for increasing semiclassical parameters, approach the results of
corresponding mathematical operations of Hamiltonian dynamics, this approach
is far from trivial, as remarked by Berry [5, 6]. In fact, it involves two non
commuting procedures: a semiclassical limit—that can take various forms, such
as the Planck constant $h$ tending to zero (that is clearly an oxymoron, for a
constant does not vary) or, as I shall do in this paper, studying particles of
larger and larger mass—and a long-time limit, in the time-span of both
evolutions. Then, formal correspondence must be reformulated as a question of
the scaling relations between time, a semiclassical parameter, and the
(properly formulated) difference between classical and quantum results [7].

To be sure, a particular feature of this relation was immediately realized in
the early years of the study of quantum chaos: when considering the time
evolution of a classically chaotic system, combination of the Heisenberg
uncertainty principle and the Lyapunov instability of classical motion
restricts agreement of the two dynamics to times logarithmically short in the
semiclassical parameter (the Berman-Zaslavky time, sometimes also named after
Ehrenfest) [8]. Or, reversing the terms in the equation, to achieve
coincidence over linearly increasing spans of time, an unphysical, exponential
increase of the semiclassical parameter is required. Albeit previously
realized by Chirikov, Percival, Berry, Kolovsky and others, perhaps the most
catchy representation of this fact has been exhibited by Zurek and Paz by
considering the chaotic motion of Hyperion [9, 10, 11], showing that the
Berman-Zaslavsky time for this celestial object is much too small to agree
with observations. The resolution of this paradox has been proposed invoking
the phenomenon of decoherence [12, 13, 14], which, in essence, limits the
resolution of phase-space by blurring quantum evolution—and classical as
well—via interaction with an environment. An alternative view that I do not
examine here was presented in [15]. I have discussed the historical origins of
the decoherence approach in [16, 17]: particularly relevant in my view are
references [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

Since the notion of decoherence invokes interaction of a system with its
environment, many models have been proposed to this aim. Most of them turn the
evolution of the density matrix into a master equation of Lindblad form [32].
In this paper, I keep pursuing the goal of representing this interaction
exactly, via the model of the multi-particle Arnol’d cat [16]. This model has
a fundamental difference with respect to most models: it is fully unitary and
as such it is immune to dissipation. Moreover, it avoids recourse to any
approximation, or the introduction of infinite baths of oscillators, or even
random dephasing of the quantum evolution: these are sound procedures, but
they input to the system an infinite amount of information which hinders the
possible organization of information coming from the exact Hilbert space
dynamics. To appreciate this otherwise obscure remark, the reader is referred
to the more complete discussion in [16, 17]. Furthermore, the proposed system
gains more relevance because it permits to derive derive scaling laws in terms
of the physical parameters (that shall be described momentarily) that cannot
be easily disentangled by the usual approaches mentioned above.

Let me now introduce briefly the content of this paper. The Hamiltonian of the
multi-particle Arnol’d cat, briefly described in Section 2, is that of a free
particle on a ring subject to periodic impulses (kicks in the jargon) that
change its momentum instantaneously [3], in such a way that the Floquet map of
the classical motion is the Arnol’d cat mapping [33]. Moreover, this large
particle is coupled via elastic scattering to a number of smaller particles,
also rotating in the ring [16]. This coupling and the full evolution can be
computed to any desired accuracy by a precise numerical technique. Then, we
assume that we are allowed to measure the dynamical variables of the large
particle only, which constitutes the system, while the remaining particles are
the environment, and yet we treat the two as a whole quantum system. An early
example of this procedure is Kolovsky’s treatment of two coupled standard maps
on the torus [34].

It is then important to examine the dynamics of this system in the light of
the commonly accepted indicators of dynamical complexity. In my view the most
important concept in this category is the quantum dynamical entropy [35, 36,
37], which corresponds to the Kolmogorov Sinai classical entropy [38, 39]. Its
limited appearance in the literature is most probably due to the fact that it
is extremely difficult to compute, even numerically and even in simple models
[36, 37, 40]. I studied this entropy in [40, 16] and in a previous paper in
this journal [17]. Instead, in [41] I focused attention to the exact Von
Neumann entropy (not the linear version of it) of the reduced density matrix,
obtained by tracing the degrees of freedom of the small particles. In many
investigations (see the recent [42] and references therein) the initial growth
of this quantity has been related to classical Lyapunov exponents. While
permitting to “observe” visually the decohering effect on Schrödinger cats
wave-function [43], study of the multi-particle Arnol’d cat system raises some
caveats on the above conclusions, because it shows that a careful scaling of
the physical parameters is required to achieve the desired result. In addition
to the dynamical entropy, various attempts have been performed to introduce a
quantum notion of Lyapunov exponents [44, 45, 46, 47]. They also reveal the
limitations that quantum dynamics poses to the development of a true dynamical
instability.

The aim of the present work is to continue the analysis of the multi-particle
Arnol’d cat by investigating two quantities that come in the form of the trace
of products of dynamical operators corresponding to classical canonical
variables. Again, we consider position and momentum operators $Q$ and $P$
corresponding to the large particle. The first quantity, considered in Section
3, is the time autocorrelation function (ACF for short) of the position $Q$,
$C_{Q}(t)=\mbox{Trace }Q(t)\rho Q$, where $\rho$ is an invariant density
matrix and $Q(t)$ is the time evolved operator $Q(t)=U^{-t}\;Q\;U^{t}$. Here
$U^{t},\;t\in\mathbf{R}$ is the group of quantum evolution operators.

This quantity, when $Q$ is replaced by a generical dynamical variable subject
to certain conditions, has been studied long ago [48, 49] in the case of the
kicked rotor, showing correspondence only for times logarithmically short in
the semiclassical parameter, and power-law decay with special features
afterwards. A different result was found for the evolution generated by
symmetric matrices (Hamiltonians) in the GUE ensemble [50]. It was shown that
the average of ACF’s over the statistical ensemble of matrices tends to a
constant, bottom value that depends on the size of the matrix $\cal N$ (that
is, the dimension of the Hilbert space). This value vanishes when $\cal N$
increases, and so it does the dispersion of the autocorrelation functions
generated by individual Hamiltonians. Furthermore, the initial decay is
extremely rapid, and it takes place on a time span that also vanishes when
increasing $\cal N$. In short, random matrices dynamics approaches a pure
Bernoulli type behavior [50].

In this paper, I investigate the position ACF in four dynamical situations:
when the large particle is free (i.e. not subject to periodic kicks nor to
small particle scattering), when kicks are switched on but the scattering
cross section is null, when this situation is reversed, and finally when both
kicks and scattering are present. Comparison with the random matrix result
just quoted and with the classical theory will be presented.

Summarizing the obtained results, in order, we will find that motion of the
quantum free particle corresponds to a linear transformation of the torus, in
such a way that, at any finite precision, agreement between classical and
quantum results extends for a time range that scales as the mass of the
particle: the correspondence principle is here physically vindicated. Next,
the effect of the scattering perturbation will be shown to enhance the decay
of the ACF in the free particle case, but more importantly, to quench the
periodic resurgences of the quantum Arnol’d cat in such a way to reproduce the
random matrix behavior quoted above.

In the second part of the paper, focus is on the out of time correlator
(commonly abbreviated as OTOC) of position and momentum, which is also defined
in terms of traces of product of operators: $O(t)=\mbox{Trace
}[Q(t),P]\;\rho\;[Q(t),P]^{\dagger}$, where $[\cdot,\cdot]$ is the quantum
commutator. The short time behavior of $O(t)$ has been put in relation to
classical Lyapunov exponents [51, 52, 53, 54]. I perform its analysis in the
same dynamical situations described above. The results will show that the OTOC
is constant in time for the quantum free particle, at difference with other
dynamical indicators. In this case, scattering will be shown to contribute an
increase of the OTOC that is proportional to the intensity of the
perturbation: this result is instrumental to provide a heuristic explanation
of the experimental data of the OTOC for the full dynamical system, both
before and after the Berman-Zaslavsky time scale.

In the Conclusion I briefly comment on the significance of the obtained
results.

##  2 Review of the multi-particle Arnol’d cat

The multi-particle Arnol’d cat is a generalization of the classical Arnol’d
cat when viewed as a particle freely rotating in a ring and subject to a
periodic impulsive force. It is obtained by adding a number of additional
particles in the ring, elastically scattering with the former. This
construction accomplishes the main goal of rendering the Joos–Zeh approach to
decoherence [20] in a fully unitary and controllable way. In fact, in so doing
the introduction of an infinite amount of information provided by infinite
baths of oscillator is avoided, as well as dissipative effects present in
other models: see [16] and a previous paper in this journal [17] for further
comments on this topic.

The Hamiltonian of a particle of mass $M$ that we consider is

$H_{cat}(P,Q,t)=\frac{P^{2}}{2M}-\eta\frac{Q^{2}}{2}\sum_{j=-\infty}^{\infty}\delta(t/T-j),$
(1)

where the variables $Q$ and $P$ take value in the unit torus and $T$ is the
period between kicks of amplitude $\eta$. If parameters are chosen as we will
do momentarily, the classical evolution generated by this Hamiltonian over
time $T$ is the Arnol’d cat map. Notice that classical chaos in generated by
the inverted harmonic potential in eq. (1), for $\eta>0$. To the contrary,
when $\eta<0$ the Hamiltonian generated an elliptic map, whose behavior is
more akin (i.e stable) to the free motion $\eta=0$ discussed later in this
paper.

Quantization of this and of similar systems have been studied in the last
century [55, 56, 57, 58, 3] and can be effected by a combination of kinematics
and dynamics on the space of square summable functions periodic in both
position and momentum [59, 60]. The particular Hamiltonian eq. (1) has been
quantized in this way in [3], successive to the original treatment by
semiclassical means in [58]. The canonical formalism also permits to write a
convenient finite-dimensional Wigner function [58, 3]. This approach has been
fruitfully applied (and some formulae rediscovered) in many successive works.

In short, the Hilbert space of the one-particle system eq. (1) is unitarily
equivalent to $\mathbf{C}^{\cal{\cal N}}$, where the dimension ${\cal{\cal
N}}$ is proportional to the mass $M$ of the particle, which is free to be
increased, to enter the semi-classical region, while the Planck constant $h$
is kept constant, as it should:

$\frac{ML^{2}}{T}={\cal N}h.$ (2)

In the above, $L=1$ is the periodicity of the torus in the $Q$ direction and
$T$ is the time interval between kicks, also set to unity: see [3, 16] for
more detail. To regain the classical Arnol’d cat also the dimensional
paramenter $\eta$ in eq. (1) is given the value one. It is nonetheless
important that physical quantities carry their correct dimensions. Remark that
because of eq. (2) the mass $M$ of the particle is quantized.

Evolution in this space is effected by a finite, unitary matrix $U$. Following
[3] the matrix components of $U$ in the position representation are as
follows. The free evolution operator $U^{free}$ has matrix elements

$U^{free}_{kl}=\frac{1}{\sqrt{\cal N}}e^{-(\pi il^{2}/{\cal N})}e^{2\pi
ikl/{\cal N}},$ (3)

while the kick operator $K$ is

$K_{kl}=\frac{1}{\sqrt{\cal N}}e^{i\pi l^{2}/{\cal N}}\delta_{k,l},$ (4)

where the indices $k,l$ range from $0$ to ${\cal N}-1$, $\delta_{k,l}$ is the
Kronecker delta, and finally $U=KU^{free}$.

Perhaps the main advantage of the canonical formalism over the semiclassical
is that it allows a straightforward extension of the model by adding a number
$I$ of freely rotating, smaller particles of mass $m$ on the ring, scattering
with the former via a point interaction $V(\cdot)$ that is unity when its
argument is null and zero otherwise. The Hamiltonian of the full system [16]
is therefore

$H=H_{cat}(Q,P,t)+\sum_{i=1}^{I}\frac{p_{i}^{2}}{2m}+\kappa\sum_{i=1}^{I}V(q_{i}-Q).$
(5)

The constant $\kappa$ is a sort of cross section for the scattering of the
large particle and the smaller ones.

Quantum kinematics for the small particles is formally achieved in the same
lines as for the large: In addition to the $L=1$ periodicity in the variables
$q_{i}$, we also impose periodicity $\frac{mL}{T}$ in the momenta $p_{i}$. We
obtain that the mass of the small particles is also quantized:

$\frac{mL^{2}}{T}={\nu}h,$ (6)

where $\nu$ is an integer. For convenience we choose both $\nu$ and $\cal N$
to be powers of two. Furthermore, being the allowed position of large and
small particles restricted to a lattice, we stipulate that each coarser
lattice (for the small particles) is a subset of that of the large: the
allowed positions are therefore

$(Q,q_{1},\ldots,q_{I})=(\frac{j_{0}}{\cal
N},\frac{j_{1}}{\nu}+\frac{s_{1}}{\cal
N},\ldots,\frac{j_{I}}{\nu}+\frac{s_{I}}{\cal N}),$ (7)

in which $j_{0}$ ranges from $0$ to ${\cal N}-1$ and the index $i=0$ is
related to the particle of mass $M$; $j_{i}$ range from $0$ to $\nu-1$ and
labels the position of a small particle, $i$ ranging from $1$ to $I$; finally
the integer $s_{i}$ gives the shift of the origin of the $i$-th lattice with
respect to the zeroth. Terms at the right hand side of eq. (7) are meant to be
modulus one.

The above expression also provides a basis for the Hilbert space of the
system, in the position representation, as a set of delta functions at the
positions in eq. (7). This space is therefore isometric to a power of
$\mathbf{C}$:

${\cal H}={\cal H}_{cat}\otimes{\cal H}_{small}^{I}={\mathbf{C}}^{{\cal
N}\nu^{I}}.$ (8)

According to the choice that $V$ be a local interaction, in the position
representation its matrix elements are

$V_{j_{0},j_{1},\ldots,j_{I};j^{\prime}_{0},j^{\prime}_{1},\ldots,j^{\prime}_{I}}=\prod_{i=0}^{I}\delta_{j_{i},j^{\prime}_{i}}\sum_{i=1}^{I}\delta_{j_{0},\frac{\cal
N}{\nu}j_{i}+s_{i}}.$ (9)

As explained in [16], the unitary evolution in this ${\cal N}\times\nu^{I}$
dimensional space is effected by a Trotter-Suzuki product technique. For the
computations in this paper we employ the second order approximation version.
These computations can be carried on to any desired degree of accuracy.

To define observable quantities, following standard usage we replace $Q$ and
$P$ with periodicized quantum operators on the two-torus, $Q^{\mbox{\tiny
per}}=\sin(2\pi Q)$ and $P^{\mbox{\tiny per}}=\sin(2\pi P)$. These operators
are diagonal in the position and momentum representation, respectively. It is
important to remark that they act on the coordinates of the large particle
only, so that the corresponding diagonal elements are

$Q^{\mbox{\tiny per}}_{jj}=\sin\frac{2\pi j}{\cal N},\;\;P^{\mbox{\tiny
per}}_{kk}=-\sin\frac{2\pi k}{\cal N},$ (10)

where $j$ and $k$, ranging from $0$ to ${\cal N}-1$, label position and
momentum of the large particle in the corresponding representations. For
easiness of notation, in the following we drop the periodicity superscript.
Unitary transformation between the two representations is performed by a
discrete Fourier transform conveniently coded by a fast Fourier numerical
scheme (which must obviously comprise the tensor product Hilbert space in eq.
(8)). Computations are faster when choosing $\cal N$ to be a power of two, as
obvious.

##  3 The autocorrelation function of the position

As described in the introduction, to investigate the dynamics of the multi-
particle Arnol’d cat we first consider the time autocorrelation function of
the position, defined as

$C_{Q}(t)=\langle Q(t)\;\rho\;Q\rangle,$ (11)

where $Q$ is the periodicized position operator acting only on the large
particle variable, eq. (10). In terms of the evolution operator, assuming
$\rho$ to be the identity, divided by the dimension of the Hilbert space, so
to correspond to the Lebesgue measure on the torus that is classically
invariant, this becomes

$C_{Q}(t)=\langle U^{-t}\;Q\;U^{t}Q\rangle,$ (12)

where here and in the following $\langle\cdot\rangle$ indicates the trace
divided by the Hilbert space dimension.

###  3.1 ACF for the free particle

We first study the function $C_{Q}(t)$ (we drop the subscript $Q$ for
conciseness) when the large particle is alone ($I=0$) and not subject to the
Arnol’d kick ($\eta=0$). The classical dynamics on the two-torus is given by
the invertible map

$\Phi^{t}(Q,P)=(Q+Pt,P)\;\mbox{ mod }1,$ (13)

The correlation function of the periodic position is given by the integral

$C_{\mbox{\tiny cl}}(t)=\int\\!\int sin(2\pi Q(t))\sin(2\pi Q)\;dQ\;dP,$ (14)

which can be easily integrated to show that the classical ACF is given by half
the function sinus cardinal:

$C_{\mbox{\tiny cl}}(t)=\frac{\sin(2\pi t)}{4\pi t}=\frac{1}{2}\mbox{
sinc}(2\pi t).$ (15)

As it is apparent, this function features periodic oscillations mitigated by a
slow, power law decay. In the following figures we find it more illustrative
to plot the value of ACF multiplied by two, since the average value of
$\sin^{2}(2\pi Q)$ is one half. In so doing the ACF at time zero takes the
value one.

Next, we compute the quantum ACF for finite ${\cal N}$ and $\kappa=0$ and
compare it with the classical value eq. (15). Firstly, it is to be remarked
that the quantum ACF cannot feature the same decay as the classical, since it
is a periodic function of time. In fact, let us consider the free evolution
over an integer span of time $t$. According to the choice of dimensional
constants presented in Section 2, which leads to eq. (13), the motion is
simply given by $Q(t)=P(0)t+Q(0)$, $P(t)=P(0)$. As mentioned before, the
evolution of the Wigner function at integer times is governed by the classical
map on a ${\cal N}\times{\cal N}$ lattice, because of a commutative diagram
permitting to retrieve one from the other (see e.g. eq. (37) in [3]), with the
sole difference of a phase factor. As a consequence, when $t$ is a multiple of
${\cal N}$, the quantum motion retrieves the initial condition (modulus a
phase that can affect the sign of the ACF) and forcefully cannot follow the
classical decay in eq. (15), as seen in figure 1.

Figure 1: Position ACF $C(t)$ for the quantum free particle when $M=2^{5}h$.
The anti-periodicity at $t=32$ is evident.

Nonetheless, quantum motion does approximate the classical better and better
and for longer time spans as the mass $M$ and therefore ${\cal N}$ grow. While
this analysis can be performed analytically to a certain extent, we prefer to
carry it out numerically, also to better compare its results with those of the
next sections, when we will compute the ACF in the case of the multi-particle
Arnol’d cat.

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Figure 2: Autocorrelation function of position $C(t)$ (left vertical scale)
for the free particle of mass $M=2^{7}h$ (green crosses), $M=2^{8}h$ (blue
stars) and $M=2^{9}h$ (red open squares). The continuous black line is the
classical correlation function $C_{\mbox{\tiny cl}}(t)$ in eq. (15). Also
plotted as vertical segments is the difference $\Delta C(t)$ between quantum
and classical ACF (right vertical scale), same color coding of the symbols as
before. In the left frame the time interval $[0,1]$ is plotted, while in the
right frame $t$ belongs to $[8,10]$. Notice that as time increases, $C(t)$
decays, but the difference $\Delta C(t)$ does not.

We therefore consider a time span shorter than the periodicity $t={\cal
N}=M/h$. In figure 2 we plot $C_{\mbox{\tiny cl}}(t)$, the quantum $C(t)$ and
the difference $\Delta C(t)=C(t)-C_{\mbox{\tiny cl}}(t)$ for increasing values
of $M$ and two time windows: the first (left panel) at the beginning of the
evolution, the second between $t=16$ and $t=20$. We observe that, as expected,
$\Delta C(t)$ diminishes as $M$ grows, while—albeit oscillating—it is roughly
constant or at most slowly increasing in time. To quantify the validity of
this observation in figure 3 we compute $\Sigma\Delta C(t)$, which is the
integral of $|\Delta C(s)|$ from $s=0$ to $s=t$, when ${\cal
N}=2^{6},2^{7},2^{8},2^{9}$. These values are indicative of an average linear
increase of $\Sigma\Delta C(t)$ with time. Moreover, when plotting the
rescaled quantity ${\cal N}\times\Sigma\Delta C(t)$ as in the upper part of
figure 3, we find that the different curves collapse on a single law of the
kind $h(t)=At$. This leads to the conclusion that

$\Delta C(t)\sim\frac{Ah}{M}+o(1),\;\mbox{ for }t\ll\frac{M}{h}$ (16)

where $A$ is a constant independent of ${M}/{h}$.

Suppose now that we fix an upper bound $\delta$ to $\Delta C(t)$. The mass $M$
required to satisfy this bound must grow linearly in $\frac{1}{\delta h}$, and
the time-span of agreement grows linearly with $\frac{M}{h}$. This is the
well–known conclusion holding for integrable or almost integrable systems:
classical limit of quantum mechanics can be achieved with a physically sound
increase of the semiclassical parameter.

Figure 3: ACF of the large particle’s position in the free particle case.
Integral of $|\Delta C(s)|$ from $s=0$ to $s=t$ for $M=2^{6}h,\cdots,2^{9}h$.
The integral has not been normalized (i.e. shown is an arbitrary multiple of
the integral, the same for all data sets). Fitting lines are a linear
dependence. The top sets of data are the rescaled points $M/h$ times
$\Sigma\Delta C(t)$, running parallel to the line $h(t)=t/10$. See text for
discussion.

###  3.2 ACF for free quantum motion in the presence of scattering

We now introduce the effect of scattering on the motion of the free quantum
particle, i.e. in the absence of the cat kick ($\eta=0$). In this case the ACF
is a quasi-periodic function of time, but due to the exponentially increasing
dimension of the Hilbert space with the number of small particles, we expect
periodicities or almost periodicities to manifest themselves far in the time
evolution. Actually, the dimension count in eq. (8) is one of the original
motivations behind the introduction of this system.

In figure 4 we plot the autocorrelation function $C(t)$ in various dynamical
situations, characterized by the same value of $M$ and $m$ and the same number
of small particles (therefore, the same “quantum vs. classical” situation),
but increasing coupling $\kappa$. Also plotted is the classical value
$C_{\mbox{\tiny cl}}(t)$ (which can be thought of as the $M=\infty$ case). The
two panels in the figure show the correlation function at the beginning of the
evolution and after ten cycles of the unperturbed motion. Observe the
different vertical scale in the two panels. An increased decay of the
correlation function for increasing perturbation by the small particles is
observed.

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Figure 4: Autocorrelation function $C(t)$ of the large particle’s position in
the presence of scattering in various dynamical situations, as detailed in the
legend. In the left frame the time the interval $[1,3]$ is plotted, while in
the right frame $t$ belongs to the interval $[10,12]$.

To better gauge this effect we compute $\Sigma|C(t)|$, the integral of the
absolute value of the correlation function up to time $t$

$\Sigma|C(t)|=\int_{0}^{t}|C(s)|\;ds.$ (17)

In the case of the free particle of infinite mass (that is, in classical
dynamics) this function is logarithmically divergent—a first year calculus
exercise. Therefore, when plotted versus the logarithm of time, it features a
linear graph. In figure 5 the same law appears to characterize—in the
investigated range—the free rotation of mass $M=2^{8}h$ already described in
figure 2. To the contrary, when coupling is effective, the correlation decay
observed in figure 4 is revealed by the slower increase of $\Sigma|C(t)|$,
suggesting that in this range this function can be parameterized as
$S(t)=A_{\kappa}(1-B_{\kappa}t^{1-d_{\kappa}})$ with $d_{\kappa}$,
$A_{\kappa}$ and $B_{\kappa}$ suitable constants. This functional form applied
to the case when three particles of mass $2h$ collide with the large particle
of mass $M=2^{8}h$ yields a good fit with $d_{\kappa}\simeq 1.66$, when
$\kappa=64$ (shown in the figure). Similar data sets in figure 5 are fitted by
$d_{\kappa}=1.15$ for $\kappa=32$ and $d_{\kappa}=1.92$ for $\kappa=256$. As a
consequence, in this range, the ACF behaves like

$|C(t)|\sim B_{\kappa}t^{-d_{\kappa}}.$ (18)

A referee suggested to put this result in relation to the anomalous diffusion
characterizing a large family of classical systems [61, 62]. In the quantum
domain spreading of wave–packets governed by power-laws with many exponents is
the content of the so-called theory of quantum intermittency, see [63] and
references therein, that has the flavor of a spectral theory, unrelated to a
classical analogy. In the present context, though, a possible explanation of
the faster decay of correlations can be obtained by a purely classical
heuristic argument, as follows. For the totally free motion, the sinc decay
comes from the shear of layers of phase space at different values of $P$.
Collision with the additional particles has the effect of changing the
momentum of the large particle, moving its phase–space coordinates to a
different layer with different velocity, a fact that enhances the divergence
of nearby trajectories. A model for this behavior has been studied in [64], in
connection with flow in porous materials.

To summarize, the effect of the environment is to increase the exponent of the
power-law decay of the quantum ACF with respect to the unperturbed classical,
in a time range that scales as $t\ll\frac{M}{h}$, as in the previous
subsection. Presumably, the same behavior is also to be found in the classical
dynamics of the many–particle system, not studied in this paper.

Figure 5: Autocorrelation function $C(t)$ of the large particle’s position in
the presence of scattering. Plotted versus time is the (non-normalized)
integral of $|C(s)|$ from $s=0$ to $s=t$ in the dynamical situations listed in
the legend. Fitting by the function $S(t)$ described in the text has been
performed in the interval $\log(2t)\in[2.5:3.2]$. The black curve refers to
$\kappa=64$. See text for further discussion.  Figure 6:  Autocorrelation
function $C(t)$ for the multi-particle Arnol’d cat for various dynamical cases
listed in the legend. See text for discussion.

###  3.3 ACF for the multi-particle Arnol’d cat

Finally, let us include the cat kick ($\eta=1$), so that the full dynamical
system is considered. As mentioned before, the quantum Arnol’d cat is
characterized by the same periodicities of the classical cat on a ${\cal
N}\times{\cal N}$ lattice, because of the commutative diagram relating it to
the quantum evolution of the Wigner function (see e.g. eq. 37 in [3]). Yet, it
is seen that except for these periodicities, the autocorrelation function of
the quantum cat is null at integer times that are not multiple of the
period—indeed, the situation is as shown in figure 6. On the $2^{4}\times
2^{4}$ lattice the classical cat is periodic of period $\tau=12$. The
autocorrelation function is symmetric around zero and null at time one. The
behavior for $t\in[-1,1]$ is periodically repeated for
$t\in[-1+l\tau,1+l\tau]$, for $l$ integer.

Simple computations in modular arithmetic show that, when ${\cal N}=2^{j}$
like in the examples in this paper, the period $\tau_{j}$ is given by the
formula

$\tau_{j}=3\cdot 2^{j-2}\Rightarrow\tau(M)=\frac{3}{4}\frac{M}{h}$ (19)

that is, the period grows as the mass $M$. See [65, 66] for the general case
and [67] for the first picture of this phenomenon. The quantum Arnol’d cat
somehow achieves what happens for the evolution generated by random matrices,
as discussed in the Introduction, albeit it reveals its algorithmic simplicity
by these recurrences.

Coupling with the additional particles has the effect of preventing such
recurrences, as shown in figure 6. Particularly instructive is the curve
obtained for $I=3$ and $\kappa=1$. In fact, we see that while in the zero
period the curve quite approximately reproduces the $\kappa=0$ case (i.e. the
original quantum cat), in the first period the periodic peak reaches an
amplitude of about $0.8$, further diminishing to about $0.4$ in the second.
This quenching effect is greatly amplified as the intensity $\kappa$ is
augmented (curves with $\kappa=2,4,8,16$).

The multi-particle quantum cat ACF is therefore a dynamical realization of the
behavior of the ensemble averaged GUE Hamiltonians.

##  4 The out-of-order commutator of position and momentum

The correspondence principle has been employed to unveil the dynamical
importance of the out-of-order commutator between the two self-adjoint
operators $Q$ and $P$. In fact let $O(t)$ be defined as

$O(t)=\langle[Q(t),P]\;\rho\;[Q(t),P]^{\dagger}\rangle$ (20)

where $\langle\cdot\rangle$ is the trace operation in the Hilbert space of a
system, $\rho$ is a density matrix and $Q(t)=U(t)^{\dagger}QU(t)$, $U(t)$
being the unitary evolution group. Then, if $Q$ and $P$ take on the classical
interpretation as canonical position and momentum coordinates, replacing the
quantum commutator $[Q(t),P]=Q(t)P-PQ(t)$ by the classical Poisson bracket
$\\{Q(t),P\\}=\frac{\partial Q(t)}{\partial Q}\frac{\partial P}{\partial
P}-\frac{\partial Q(t)}{\partial P}\frac{\partial Q(t)}{\partial P}$,
immediately yields

$\\{Q(t),P\\}=\frac{\partial Q(t)}{\partial Q},$ (21)

which calls into play the local maximal Lyapunov exponent at $Q$:
$\\{Q(t),P\\}\sim e^{\lambda(Q)t}$. One should therefore expect that the
quantum $O(t)$ should behave accordingly, as the exponential of time
multiplied by twice the classical Lyapunov exponent.

From this heuristic remark one immediately observe that correspondence should
require $\rho$ to be a pure state $|\psi_{Q}\rangle\langle\psi_{Q}|$ of
minimal uncertainty at position $Q$. The obvious dependence on $Q$ calls into
play the well known classical large deviation theory of Lyapunov exponents. A
detailed study of the large deviation function for quantum Lyapunov exponents
has been recently published in this same journal [54]. It is also to be
remarked that different actions, such as taking the logarithm of $O(t)$,
tracing the logarithm of square commutator, or averaging the latter over
different $|\psi_{Q}\rangle\langle\psi_{Q}|$ as in [53], give different
results, as explained in [51]. In this work we do not investigate such
alternative procedures, but stick to the definition in eq. (20).

The validity of the short time relation to Lyapunov exponents has been
confirmed in a series of works (see e.g. [51, 52, 53, 54] and references
therein), all exhibiting breakdown of correspondence at the Berman-Zaslavsky
time scale. In line with the discussion in the introduction, it is therefore
natural to ask what decoherence can add to the picture. We can start from a
formal argument.

We assume that the density matrix $\rho$ is the identity divided by the
Hilbert space dimension, since this yields the natural analogue of the
invariant Lebesgue measure on the classical two-torus. Then, $O(t)$ in eq.
(20) is equal to the difference of two real quantities
$O(t)=O_{+}(t)-O_{-}(t)$, given by

$O_{+}(t)=2\;\langle P^{2}Q^{2}(t)\rangle,\;\;O_{-}(t)=2\;\langle
PQ(t)PQ(t)\rangle,$ (22)

more specifically

$O_{+}(t)=2\;\langle P^{2}\circ U^{-t}\circ Q^{2}\circ
U^{t}\rangle,\;\;O_{-}(t)=2\;\langle P\circ U^{-t}\circ Q\circ U^{t}\circ
P\circ U^{-t}\circ Q\circ U^{t}\rangle.$ (23)

The composition symbol $\circ$ in the above is superfluous, but it enhances
readability of these and successive formulae. Let us compare these relations
with the decoherence matrix $D_{\theta,\eta}$ that is central in the notion of
dynamical entropy, employed in the previous paper in this journal [17]. From
Sect. III therein we see that $D_{\theta,\eta}$ is the trace of the sequence
of operator products

$\Pi_{\theta_{0}}\circ U^{-1}\Pi_{\theta_{1}}\circ\cdots\circ
U^{-1}\Pi_{\theta_{n-1}}\circ\Pi_{\sigma_{n-1}}U\circ\cdots\circ\Pi_{\sigma_{1}}U\circ\Pi_{\sigma_{0}}$
(24)

where $\Pi_{i}$ have been chosen as projection operators for $Q$ in a
partition of the unit interval by sub-intervals of equal size, and $\theta$,
$\eta$ are symbolic words. We see that in this approach, the symbolic history
of the quantum motion is considered, at successive time intervals, while in
eq. (23) only the initial and final states of the motion contribute.

Moreover, passing from form to substance, the results of [16, 17] suggest—or
at least leave the possibility open—that the interaction of a system with the
environment could enable the former to produce information at the classical
rate beyond the Berman-Zaslavsky time, and more importantly, for a range
scaling linearly with the number of small particles representing the
environment in the model under study. It is then natural to wonder whether the
OTOC could also provide a tool to investigate this phenomenon.

###  4.1 OTOC for the free particle

It is instructive to notice that one can explicitly compute the OTOC in
certain cases. For a single free particle, both $O_{+}(t)$ and $O_{-}(t)$ take
a constant value:

$O_{+}(t)=1/2,\;\;O_{-}(t)=\frac{1}{2}\cos(2\pi/{\cal N}),$ (25)

as it easily verified by direct computation. Therefore, $O(t)$ is constant in
this case, at difference with the dynamical complexity of the classical
system, which increases logarithmically, as well as the quantum Alicky Fannes
entropy [40] and the Von Neumann entropy [41]. This fact seems to already
indicate a difference between the former indicators and the OTOC.

###  4.2 OTOC in the presence of scattering of the large particle by smaller
ones

Let us now study the effect of scattering by small particles on the OTOC of
the large particle variables, investigated via the autocorrelation function in
section 3.2.

In figure 7 we plot $O(t)$ (left panel) and $\log O(t)$ (right panel) versus
time for exponentially increasing values of the coupling $\kappa$, over a
large range of values from $\kappa=8$ to $\kappa=256$, when $M=2^{8}h$, $m=2h$
and $I=3$. We observe that $O(t)$ tends to a limit value $O(\infty)$ that
depends on $\kappa$. For the largest coupling, wide oscillations of $O(t)$ are
observed, related to a strong perturbation of the motion of the large
particle.

Let $\Delta{O}(t)$ be the increase of the logarithm of the OTOC from its
initial value, that corresponds to the constant value of the OTOC for the free
particle: $\log O(0)=\log[1-\cos(2\pi/{\cal N})]-\log(2)$, approximately equal
to:

$\log O(0)\simeq 2\log\frac{\pi}{\cal N}=-2\log\frac{M}{h\pi},$ (26)

as it follows from eq. (2). We therefore define

$\Delta{O}(t)=\log O(t)+2\log\frac{M}{h\pi}$ (27)

and we use the asymptotic value $\Delta{O}(\infty)$ to quantify the effect of
the perturbation, which is seen to grow in a way approximately proportional to
$\kappa$: the fitting line in figure 8 has slope $a\simeq 1.18$:

$\frac{O(\infty)}{O(0)}\sim\kappa^{a}.$ (28)

This information will be used later in Section 4.4, to explain the behavior of
the OTOC for the full system: scattering yields a contribution proportional to
$\kappa$ to the OTOC, which is achieved within a short time span. Observe in
fact that the time unit is of the same order as the average time that a free
particle needs to make a full turn of the ring.

|
---|---
Figure 7: OTOC function $O(t)$ for scattering of a large particle of mass
$M=2^{8}h$ by $I=3$ particles of mass $2h$, versus time, for
$\kappa=8,16,32,64,128,256$, curves ordered from bottom to top. In the right
frame $\log(O(t))$ in the same situation.  Figure 8: The asymptotic value
$\Delta{O}(\infty)$ defined in eq. (27) versus $\log(\kappa)$ for the data in
figure 7. The fitting line has slope $a=1.18$.

###  4.3 OTOC for the quantum Arnol’d cat

Let’s now turn to the one-particle quantum cat. The short time behavior of
$O(t)$ features an exponential increase with rate twice the classical Lyapunov
exponent, as well established in many previous works (see e.g. [51] and
references therein). We consider in figure 10 the natural logarithm of $O(t)$,
which at time zero takes the value in eq. (26). Albeit simply derived, this
equation once more underlines the crucial feature of quantum mechanics brought
about by Heisenberg principle: the finest resolution of dynamical variables is
dictated by ${\cal N}$, which in turn physically corresponds to the mass of a
quantum particle. In fact, observe in eq. (7) that the wave function of a
quantum particle of mass $M$ on the unit torus effectively lives on a regular
lattice of spacing $1/{\cal N}$.

Figure 9: The scaled quantity $\Delta{O}(t)$ defined in eq. (27) versus time,
for $M=2^{j}h$, $j=6,\ldots,11$, curves increasingly ordered from bottom to
top. The strait line has slope twice the Lyapunov exponent of the classical
cat.

In the context of OTOC, the value in eq. (26) corresponds to the minimal
uncertainty in the $Q$ variable in eq. (21) at time zero, which increases
exponentially immediately after. Even if repetitive of what has since long
been described in the early works on quantum chaos, and verified in many
successive papers, let us first study the behavior of this quantity when
increasing exponentially the mass of the large particle, in the absence of
scattering (i.e. when $\kappa=0$). To do this, in figure 9 we plot versus time
the quantity $\Delta{O}(t)$ defined in eq. (26). It is observed that the
curves initially coincide to high precision and the rate of exponential
increase is twice the Lyapunov exponent. This situation lasts for larger spans
of time the larger the value of the mass of the particle. Successively, they
feature the typical interference behavior of the quantum cat. The explanation
is quite simple: because of the scaling involved in eq. (27) the exponential
growth of an initial difference in position stops when it has reached the
(unit) size of the torus. This time is shorter the larger the initial
difference. Nothing new here: this is the usual explanation of the Zaslavsky-
Ehrenfest time scale.

In addition, the coincidence of the scaled curves goes in favor of the formal
validity of the classical limit: the larger the mass of the particle, the
longer the correspondence of classical and quantum results holds. Nonetheless,
recall the objection at this point: the range of this correspondence scales
unphysically with the system parameters.

###  4.4 OTOC for the multi-particle Arnol’d cat

The behavior of the OTOC seen so far is well known, see e.g. [52], albeit we
hope to have made it clear why our assessment of its physical significance is
different from the common lore. But let us now introduce coupling between the
large particle and the smaller ones. We begin to expose our results by
commenting two figures with experimental results.

Firstly, in fig. 10 we plot the logarithm of $O(t)$ for a particle of mass
$M=2^{8}$, subject to the Arnol’d kick and scattered by $I$ small particles of
mass $m=2$. We consider various combinations of $I$ and of the coupling
constant $\kappa$ that measures the intensity of the scattering. In the case
$I=0$ we recover the unperturbed quantum cat just studied.

Let us first consider the short–time behavior. We observe that the effect of
the perturbation in this logarithmic scale is more evident in the initial
moments of the evolution, while successively the curves tend to realign
themselves with the $\kappa=0$ case. The initial discrepancy is more
noticeable the larger the scattering intensity, both in terms of the number of
scattering particles $I$ and of the coupling $\kappa$.

Next, in figure 11 we plot the value of $O_{-}(t)$, at integer times, in the
same dynamical situations as in the previous figure. In all cases the time
zero value $O_{-}(0)$ is that of the single free particle,
$O_{-}(0)=\frac{1}{2}\cos(2\pi/{\cal N})$.

Figure 10: Logarithm of the OTOC function $O(t)$ for $M=2^{8}h$, $m=2h$ . The
slope of the straight dotted line is twice the Lyapunov exponent of the
classical cat. Different values of $I$ and $\kappa$ are investigated.  Figure
11: Quantity $O_{-}(t)$ for a particle of mass $M=2^{8}h$, subject to the
Arnol’d kick, and scattered by $I$ small particles of mass $m=2h$. The
scattering constant is $\kappa$. Values of $I$ and $\kappa$ are as in figure
10.

The short time behavior can be defined here by positivity of $O_{-}(t)$ (and
therefore $t\leq 4$ in the figure). In this region the effect of scattering is
to reduce the value of $O_{-}(t)$ with respect to the corresponding value with
no scattering ($I=0$ or equivalently $\kappa=0$). Since the value of
$O_{+}(t)$ is practically unaltered, this implies an increase of the value of
$O(t)$ with respect to the unperturbed cat value, observed by plotting the
logarithm of the total quantity $O(t)$ in the previous figure 10.

Figure 12: Logarithm of the OTOC function $O(t)$ for $M=2^{9}h$, $m=2h$ and
various values of $I$ and $\kappa$. See text for explanation.

We will consider the long time behavior of these quantities later on. Let us
now try to frame theoretically the previous observations. In this regard it is
helpful to display $O(t)$ also at at non–integer times. Recall that the
impulsive perturbation is taking place at integer times, so that in between
kicks the dynamics of the large particle is only influenced by scattering by
the smaller ones. In figure 12 we repeat the analysis of figure 10, now for a
larger mass $M=2^{9}h$. For comparison, we plot again in the same figure the
OTOC for the case $I=3$, $\kappa=64$ in the absence of the cat kick, that is,
$\eta=0$ in eq. (1). Pay attention to the data for $\kappa=64$. We observe
that the increase in the logarithm of OTOC taking place in between
kicks—therefore brought about by scattering alone—diminishes with time. When
passing to a linear scale (not displayed) the increase of OTOC between kicks
is approximately constant.

The analysis of section 4.2 can now be used to propose a mechanism for these
dynamics along the following lines. Firstly, scattering of the large particle
by smaller ones produces a fixed perturbation, in the unit time between kicks.
Secondly, this perturbation adds to the divergence of trajectories, that is,
to the value of $O(t)$ during each period between kicks. Thirdly, while this
contribution is noticeable for short times, for larger times it is dwarfed by
the exponential increase of $O(t)$ entailed by the short-term chaotic
behavior. This explains the behavior observed in fig. 10. In the conclusion we
will comment on the relation of this result with other theoretical approaches.

The above analysis can be made quantitative. We define the ratio of the
quantity $O_{-}(t)$ for $I$ and $\kappa$ greater than zero with respect to the
unperturbed quantum cat:

$\rho_{\kappa}(t)=\frac{O_{-}(\kappa=0;t)}{O_{-}(\kappa;t)}.$ (29)

For vanishing values of the coupling $\kappa$ the ratio $\rho_{\kappa}$ tends
to one. Therefore, we find it convenient to consider the logarithm of
$1-\rho_{\kappa}(t)$ as a function of time. This quantity is plotted in figure
13, for $M=2^{8}h$, $m=2h$, $I=3$ and different values of $\kappa$,
$\kappa=2^{j}$, with $j=2,\ldots,7$.

Figure 13:  Logarithm of $1-\rho_{\kappa}(t)$ versus time, where
$\rho_{\kappa}(t)$ is defined in eq. (29). Parameters are $M=2^{8}h$, $m=2h$,
$I=3$. The straight line has slope one half.

This numerical analysis reveals that, for small values of time,
$1-\rho_{\kappa}(t)$ grows exponentially with an exponent very close to one
half and independent of $\kappa$ over a large span of values:

$\rho_{\kappa}(t)\simeq 1-\exp\\{-C+A(\kappa)+t/2\\},$ (30)

where $C$ is a constant independent of $t$ and $\kappa$. Furthermore, the term
$A(\kappa)$ scales approximately as $\kappa$ to a power slightly larger than
one, although the data do not allow to determine a precise value (in the data,
the exponent is roughly equal to $1.3$). This provides a quantitative
explanation to the behavior of the logarithm of $O(t)$. In fact, let
$B=\frac{1}{2}$, which is the constant value of $O_{+}(t)$. Let also
$\delta(\kappa,t)=\exp\\{-C+A(\kappa)+t/2\\}$, and observe that in the initial
range of the data in figure 12 it holds that $\delta(\kappa,t)\ll 1$. Then,
simple manipulations using the behavior of $O(t)$ in the absence of scattering
yield

$\log{O(\kappa;t)}=\log{O(0;t)}+\log\\{1+\delta(\kappa,t)\;[B(\frac{\cal
N}{\pi})^{2}e^{-2\lambda t}-1]\\},$ (31)

where $\lambda$ is the classical Lyapunov exponent. At fixed time $t$ the
$\kappa$ dependence of $\delta(\kappa,t)$ implies that $\log O(\kappa;t)$
increases with $\kappa$. Next, consider the argument of the second logarithm
at r.h.s. in eq. (31). It can be rewritten as

$\mbox{arg log}=1+\exp\\{-C+A(\kappa)\\}[B(\frac{\cal
N}{\pi})^{2}e^{-(2\lambda-1/2)t}-e^{t/2}].$ (32)

Let now $\kappa$ be constant. For small times the first term in the square
bracket in eq. (32) is much larger than the second, so that when time grows
the square bracket decreases and so does the difference
$\log{O(\kappa;t)}-\log{O(0;t)}$. This fully explains the short time behavior
observed in figure 12.

Turning finally to the behavior beyond the Berman-Zaslavsky time, one sees
clearly in figures 10 and 11 that the effect of scattering is to quench the
oscillations in the OTOC that are typical of the Arnol’d cat. This result can
also be obtained by a modification of the one–particle quantum map [52].

##  5 Conclusions

The multi-particle Arnol’d cat is a simple yet complete system that I have
introduced to analyze the nature and relevance of the decoherence approach to
quantum chaos and to the correspondence principle. Based on this model, one
can test various aspects of the dynamical behavior of quantum systems with a
classical analogue. In particular, quantum dynamical entropies, as proposed by
Alicky and Fannes, [35, 37], were investigated in [40, 16, 17] and the Von
Neumann entropy in [41]. In this paper I have studied the autocorrelation
function of the position [48, 49] and the OTOC, a quantity that has recently
attracted considerable interest. Four dynamical situations have been
investigated in both cases.

The first is that of a free particle. In this situation the quantum ACF well
approximates the classical, in the sense that given a prescribed degree of
accuracy, the time span in which the two dynamics agree scales in a physically
acceptable way in the system parameters. In turn, the OTOC is here constant.

When scattering of the large particle with the smaller ones is considered, the
ACF features an enhanced power–law decay with exponent larger than one, again
for a physically acceptable range of times. Now the OTOC initially grows and
saturates at a value approximately proportional to the scattering intensity.

Next, I have considered the standard quantum Arnol’d cat that has by now been
thoroughly investigated in exceedingly many works. Here, the ACF abruptly
decays to zero, but with resurgences at periodic instants. In this case, the
OTOC shows a previously well known scaling behavior that once more highlights
the relevance of the Berman-Zaslavsy’s time scale. No new feature of this
phenomenon is offered: it is presented to highlight the theoretical concepts
in which I frame this result. To further clarify this scaling approach, first
presented in [7], I refer the interested reader to an analogous family of
phenomena occurring in planar classical billiards [68].

Finally, the full dynamics of the multi-particle Arnol’d cat has been
considered. It was shown that by suppressing periodicities decoherence it
leads to an ACF behavior of the same kind as that of random matrices in the
GUE ensemble [50]. In regards to the OTOC, decoherence affects the short time
behavior of the OTOC and the parameter dependence of this phenomenon,
heuristically and numerically estimated, confirms the following theoretical
picture, that can be taken as a concise summary.

Quantum classical relation is best considered from a Wigner function
perspective. In the family of systems that comprises the Arnol’d cat, this
perspective yields a classical motion on a square lattice, see [58, 3].
Leveraging on this fact a means to model quantum dynamics of the kicked rotor
on the torus in the presence of external perturbation has been proposed [24,
69], invoking random perturbations of the motion. The multiparticle Arnol’d
cat described in this paper achieves this fact dynamically. Via its parameter
dependence, it is possible to analyze physically the content of the
Correspondence Principle as described in the Dirac’s quote in the introduction
without recourse to any approximation. In this respect two indicators have
been studied numerically herein, the autocorrelation function, which has a
long history, and the more recent out-of-time correlator. A scaling picture of
their behavior has numerically emerged, which must be now confirmed by a more
rigorous investigation that should extend to arbitrary, but finite, number of
particles and their mass.

##  6 Acknowledgements

I would like to thank the three unknown referees. They provided valuable
comments to better the readability of this paper throughout the whole
manuscript and to focus on certain theoretical points beyond the mere
presentation of results.

Computations for this paper were performed on the Zefiro cluster at the INFN
computer center in Pisa and on the Aramis and Porthos clusters of the
Department of Science and High Technology of the University of Insubria at
Como. Enrico Mazzoni and Mario Oriani are warmly thanked for assistance.

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