# Quantum Entropies and Decoherence for 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

I study the scaling behavior in the physical parameters of dynamical
entropies, classical and quantum, in a model of collision induced decoherence
in a chaotic system. The treatment is fully canonical and no approximations
are involved or infinite limits taken. I present this model in order to
clarify my views on the debate about the nature, definition, and relevance of
quantum chaos.

Dedicated to Giulio Casati on his eightieth

##  1 Preface: Forty years with Giulio Casati

Forty years ago, in December 1982, I defended my master thesis in Pavia under
the guidance of Italo Guarneri. He had introduced me to his coworker from the
university of Milano, Giulio Casati, who in the same days was turning forty.
Ever since, my scientific career has been marked by this encounter, in
repeated periods of proximity and elongation. After a brief permanence in
Milano, Giulio managed to put me in contact with Joe Ford, with whom I was a
postdoc in Atlanta for many years, before taking a three-years turn to France
with Daniel Bessis, a fourth important figure in my education.

But now back to Giulio: while I was in Paris, using his unrivalled political
ability, he was setting up nothing less than a new university, in Como, where
I later joined him (Figure 1) and Italo Guarneri in what at the beginning was
a heroic endeavor: we shared a single office with a modem by which I connected
to computers in Paris and Milano. We worked hard to establish the course in
physics—later also in mathematics—and we brought an intense activity to the
newly instituted Center for Non-linear and Complex Systems. We organized
conferences, workshops; we invited many visitors, hosted postdocs and graduate
students. Focus of this activity was chaos in classical and quantum systems,
and also orthogonal polynomials of singular measures and their relevance for
the phenomenon of quantum intermittency [1], on which I will not dwell here.

Figure 1: Como, early 90’s. Picture by Felix Izrailev Figure 2: Joe Ford and
Boris Chirikov. Como, early 90’s. Picture by Felix Izrailev

In this paper I will instead present a reflection on the problem of quantum
chaos and the quantum–classical correspondence. Notwithstanding the dedicatory
goal, I will not hide the differences between my views and those of Giulio [2,
3, 4, 5, 6], the extremist of the middle, as Joe Ford jokingly called him, not
without a hint of affection. The nickname implied that such views, Giulio’s,
were widely shared. Indeed, to the question Does chaos pose a problem to
quantum mechanics? few would answer in the positive, and many would quote the
notions of quantum chaology [7, 8] (Berry) or quantum pseudo-chaos [9, 10]
(Chirikov). Joe Ford was convinced of the contrary and with some differences I
shared his views [11] (but compare the different perspective of [12] and
[13]). This celebration is now the occasion for me to put on the record what I
have learnt in these years and what I believe is still to be investigated.

##  2 Introduction and Summary of the Paper.

As I mentioned, Giulio introduced me to Joe Ford, who had learnt the basic
tenets of algorithmic complexity theory [28] from his colleague and friend
Boris Chirikov (Figure 2) from the Soviet Union, where this theory had
flourished thanks to the school of Kolmogorov. It leaves a sour taste to
remember the many liaisons that ensued, in a moment when we fear for many dear
friends and colleagues. This theory quantifies the amount of information
contained in a sequence of symbols [29, 30, 31, 32]. Now, symbolic dynamics is
a tool to translate motions into sequence of symbols, in such a way that one
can equate chaotic trajectories of a system to random symbolic
sequences—roughly speaking, sequences that cannot be significantly compressed
without losing information. In this perspective, chaos is the uninterrupted
production of information by the motion of a dynamical system, just in the
same way as a stochastic process could do; hence, in Ford’s wording [36],
chaos is deterministic randomness. In a historical/theoretical introduction to
classical chaos coauthored with Giulio for the Journal of the Italian Physical
Society [27] we adopted this perspective: that instability of trajectories,
transverse homoclinic intersections, horseshoes are the dynamical mechanisms
by which chaos arises, but not its essence, which is the algorithmic
complexity of symbolic trajectories.

Similarly, in quantum dynamics, a variety of indicators have been introduced
to unveil the mechanisms behind instability–or its lack. It is clearly not my
aim here to list all contributions, early or recent, to this research, but
just to name a few, for the sake of argument. Phase space representations,
like the Wigner function, have been employed since long for comparison with
classical dynamics, as in [62, 63]. In [11] this analysis was instrumental to
find the eigenfunctions of the evolution operator and to expose the
exponential propagation of mode number in the classical Liouville evolution,
which impairs stability and practical reversibility, a technique later
revisited by Benenti and Casati [2]. Quantum Lyapunov exponents have also been
defined, see e.g. in [14, 15] and more recently much attention has been
devoted to the out of time autocorrelation functions, see the paper by Kurchan
in this volume [16] and references therein.

At a more general level than these mechanisms there is the study of the
algorithmic complexity of the quantum motion. Many years ago, Berry and Hannay
quantized the Arnol’d cat map [34] via semiclassical means [35]. Ford, myself
and Ristow [11] developed a fully Hamiltonian quantization and set out to
estimate the algorithmic complexity of the evolution, finding that it grows
like the logarithm of the dimension of the Hilbert space of the system plus
the logarithm of the time–length of the evolution. Prior to saturation of such
upper bounds (see the description in Sect. 4) the quantum behavior reproduces
the classical, as it has been repeatedly reported using the other indicators
mentioned above. Yet, it was recognised that the scaling of the range of such
correspondence with the physical parameters is fully unsatisfactory to justify
a sensible approach to the classical limit (a sort of logarithmic catastrophe
[12]).

As it is well known, a possible solution to this problem has been proposed, at
the end of the last century, [17, 18, 19, 20, 21, 22, 23, 24] by considering
that systems are coupled to an external environment. I believe that this
approach still need to be completed, or at least polished, in two respects.
Firstly, by casting it in a fully Hamiltonian approach, as I shall do by
introducing a model system of collision-induced decoherence that avoids
recurse to mean field approximations or infinite baths of oscillators, by
which non-dynamical randomness might creep in uncontrolled. Secondly, by
framing it into an information theoretical perspective, employing the
so–called Alicky-Fannes (AF) entropy [25, 26], which offers computational
advantages and avoids some of the difficulties of algorithmic complexity. It
must be noted that quantum dynamical entropies, generalizing the classical KS
notion [37] were already developed in the early days of “quantum chaos” [38,
39, 40].

The scheme of this paper is as follows: in the next section I briefly recall
the notion of dynamical entropy. Section 4 discusses the importance of
dynamical entropies in the notion of chaos and the fundamental problem that
one encounters in taking the classical limit of chaotic quantum systems.
Section 5 is a review of the quantum version of the classical Arnol’d cat map
[34], as the motion of a single kicked particle, so to open the way to a multi
particle generalization. The Alicki-Fannes entropy of such single particle
system has been computed in a series of papers. The formalism is presented in
Section 6, again in view of its generalization to the multiparticle Arnol’d
cat [41], described in Section 7. This generalization aims at introducing in a
fully controllable, dynamical way the concept of decoherence induced by
scattering by light particles, following the paradigm of Joos and Zeh [42]. In
a small aside, Section 8, I briefly describe recent results on the Von Neumann
entropy of the multi-particle Arnol’d cat [43], which confirm its suitability
as model of quantum decoherence. In Section 9 I return to the AF entropy,
which in my view is theoretically more relevant than Von Neumann’s, to the
same degree as it is harder to treat theoretically and compute numerically. I
explain how a combined limit in the number of particles and their mass could
provide a physically sound means to understand the classical limit. Final
comments are presented in the conclusions.

##  3 Dynamical Entropy, Classical and Quantum

Consider a classical dynamical system [34, 37] consisting of a phase-space $X$
on which a transformation $\phi$ acts in such a way to preserve a measure
$\mu$ defined on a suitable sigma-algebra of subsets of $X$:
$\mu(\phi^{-1}(A))=\mu(A)$ for any $A$ in the sigma-algebra. The map $\phi$
generates the dynamics, that we assume to evolve in discrete time instants,
denoted by the symbol $n$ throughout the paper. Consider a finite partition
$\cal P$ of $X$ by measurable subsets $P_{l}$, $l=0,\ldots,L-1$ ,
$\bigcup_{l}P_{l}=X$. The Shannon entropy of this partition, $S({\cal P})$ is
defined as

$S({\cal P})=-\sum_{l=0}^{L-1}\mu(P_{l})\log\mu(P_{l}).$ (1)

This value is the average amount of information obtained by determining the
element of the partition that contains a given point in phase space: this
action can be physically interpreted as the result of a measurement
experiment.

Next, consider the partition obtained by intersecting all elements of ${\cal
P}$ and all their pre-images by the map $\phi$: indicate this new partition by

${\cal P}\vee\phi^{-1}({\cal P})=\\{P\cap\phi^{-1}(Q),\mbox{ for }P,Q\in{\cal
P}\\},$ (2)

where obviously some of the intersections at right hand side may be empty. If
we label sets by the alphabet composed of the symbols $0$ to $L-1$, as in eq.
(1), the elements of the partition in eq. (2) are labeled as
$P_{lk}=P_{l}\cap\phi^{-1}(P_{k})$. These sets comprise all points that at
time zero are located in $P_{l}$ and at the successive time instant in
$P_{k}$. Therefore, the Shannon entropy of ${\cal P}\vee\phi^{-1}({\cal P})$
measures the average amount of information provided by knowledge of the
symbolic dynamics of a point over two time instants.

In the lore, a word is a sequence of symbols
$\mathbf{\sigma}=\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}$ in the alphabet.
We use the notation $|\sigma|$ for the length of a word. Let $\Sigma_{n}$ be
the set of all words of length $n$ in the alphabet. Clearly, the cardinality
of this set is $L^{n}$. Every word defines a cylinder $P_{\mathbf{\sigma}}$ in
$X$: this is the set of all points $x$ for which $\phi^{i}(x)\in
P_{\sigma_{i}}$, for $i=0,\ldots,n-1$. The set of cylinders associated with
words of length $n$ compose the partition $\vee_{i=0}^{n-1}\phi^{-i}({\cal
P})$, whose entropy is therefore

$S({\cal P},n)=S(\vee_{i=0}^{n-1}\phi^{-i}({\cal
P}))=-\sum_{\mathbf{\sigma}\in\Sigma_{n}}\mu(P_{\mathbf{\sigma}})\log(\mu(P_{\mathbf{\sigma}})).$
(3)

Consider the difference $S({\cal P},n+1)-S({\cal P},n)$. This quantifies the
average amount of new information obtained by observing the state of the
motion at time $n$, on top of what already obtained by the observations at the
previous $n$ instants, from time $0$ to time $n-1$. It could also be described
as the surprise that the dynamical evolution of the system is capable of
producing at such time. It is then understandable that one can define as
dynamical entropy the asymptotic rate of information production as

$h({\cal P})=\lim_{n\to\infty}S({\cal P},n+1)-S({\cal
P},n)=\lim_{n\to\infty}\frac{1}{n}{S({\cal P},n)}.$ (4)

This leads to the definition of the Kolmogorov Sinai entropy of the dynamical
system as a supremum over partitions:

$h_{KS}(X,\phi,\mu,{\cal A})=\sup\\{h({\cal P}),{\cal P}\mbox{ finite
measurable partition of }X\\}.$ (5)

Chaotic classical dynamical systems are then properly defined as those having
positive KS entropy, and Brudno’s theorem [33] links this concept to positive
algorithmic complexity.

Let us now turn to quantum dynamics for comparison. Robert Alicki and Mark
Fannes have constructed a definition of entropy following the same steps as in
the classical construction [25, 26, 67]. A quantum dynamical system is
composed of a Hilbert space ${\cal H}$ and of a unitary evolution $U$ that
leaves invariant a density matrix $\rho$. The analogue of the partition ${\cal
P}$ is a finite family of orthogonal projection operators $P_{l}$, from $l=0$
to $L-1$, such that $\sum_{l}P_{l}=I$, $I$ being the identity. We use the same
symbol $P_{l}$ for both the classical and the quantum case, since this
highlights the analogy.

For illustration, suppose that $\rho=|\psi\\!><\\!\psi|$, even if generically
this is not invariant. The wave function $|\psi\\!>$ represents the initial
state of the system. Projecting $|\psi\\!>$ by $P_{l}$ is the analogue of the
classical measurement of the partition element which the dynamical state
belongs to. In quantum dynamics, the square modulus $\|P_{l}|\psi>\|^{2}$ is
the probability that the result of this measurement yields the value $l$.
Clearly, because of normalization, $\sum_{l}\|P_{l}|\psi\\!>\|^{2}=1$.
Following suit, the analogues of the elements of ${\cal P}\vee\phi^{-1}({\cal
P})$ are $P_{\sigma_{1}}UP_{\sigma_{0}}|\psi\\!>$, which shows that the
classical action of $\phi^{-1}$ is replaced by the quantum unitary evolution
operator $U$. This leads to define the quantum dynamical partitions

$\vee_{i=0}^{n-1}U^{i}({\cal P})=\\{P_{\sigma_{n-1}}U\circ
P_{\sigma_{n-2}}U\circ\cdots\circ P_{\sigma_{1}}U\circ
P_{\sigma_{0}},\;\mathbf{\sigma}\in\Sigma_{n}\\}=\\{P_{\sigma},\;\mathbf{\sigma}\in\Sigma_{n}\\}.$
(6)

We have used the notation $P_{\mathbf{\sigma}}$ for the operators in the above
set. Then, the quantum analogue of $\mu(P_{\sigma})$ takes the form of the
trace of a product of operators:

$\mu(P_{\sigma})=\mbox{Tr}(P^{\dagger}_{\sigma}\rho P_{\sigma}).$ (7)

Alas, it is not possible to use this quantity to define a dynamical entropy
using eq. (3). In fact, suppose again for simplicity that
$\rho=|\psi\\!><\\!\psi|$. Then

$\mu(P_{\sigma})=\|P_{\sigma}|\psi\\!>\|^{2}.$ (8)

While it is true that $\sum_{\sigma}\mu(P_{\sigma})=1$, observe that eq. (8)
can be rewritten as follows:

$\|P_{\sigma}|\psi\\!>\|^{2}=<\\!\psi P_{\sigma}|P_{\sigma}\psi\\!>.$ (9)

The word ${\mathbf{\sigma}}$ appears twice in the above equation. Replace the
first appearance by a different word ${\mathbf{\theta}}\in\Sigma_{n}$, to get

$\displaystyle<\\!\psi
P_{\theta}|P_{\sigma}\psi\\!>=<\\!\psi|P^{\dagger}_{\theta}P_{\sigma}\psi\\!>=<\\!\psi|P^{\dagger}_{\theta}P_{\sigma_{n-1}}U\circ\cdots\circ
P_{\sigma_{1}}U\circ P_{\sigma_{0}}\psi\\!>=$
$\displaystyle=<\\!\psi|P_{\theta_{0}}\circ
U^{-1}P_{\theta_{1}}\circ\cdots\circ U^{-1}P_{\theta_{n-1}}\circ
P_{\sigma_{n-1}}U\circ\cdots\circ P_{\sigma_{1}}U\circ
P_{\sigma_{0}}\psi\\!>.$

It helps intuition to read the last line from right to left: in sequence, the
initial function $|\psi\\!>$ is projected by $P_{\sigma_{0}}$, acted upon by
$U$, projected by $P_{\sigma_{1}}$, acted upon by $U$, and so on until it is
projected by $P_{\sigma_{n-1}}$. Then, the adjoint of $P_{\mathbf{\theta}}$
operates by projecting by $P_{\theta_{n-1}}$, acting by the inverse of $U$
(that is, by reversing time), projecting by $P_{\theta_{n-2}}$, acting by
$U^{-1}$ and so on until the projection by $P_{\theta_{0}}$ and finally taking
the scalar product with the initial function $|\psi\\!>$.

Clearly, the result of this operation is null unless
$\sigma_{n-1}=\theta_{n-1}$. In the classical case where $\phi^{-1}$ takes the
place of $U$ the result is also null unless the full word $\mathbf{\theta}$
coincides with $\sigma$. In fact, it is enlightening to consider the above
process as a sequence of many–slits experiments: a dynamical trajectory must
go through the slits $\sigma_{0}$ to $\sigma_{n-1}$. Then, time is reversed
and it is required to pass from the slits $\theta_{n-1}$ to $\theta_{0}$.
Since trajectories in classical dynamical systems are time reversible, this is
only possible if $\mathbf{\theta}=\mathbf{\sigma}$. On the contrary, in
quantum dynamics two different paths from $\sigma_{0}=\theta_{0}$ to
$\sigma_{n-1}=\theta_{n-1}$, the end points being fixed, may not be orthogonal
because of quantum interference: $<\\!\psi
P_{\theta}|P_{\sigma}\psi\\!>=<\\!\psi_{\theta}|\psi_{\sigma}\\!>\neq 0$,
where we have introduced the notation $|\psi_{\sigma}\\!>=P_{\sigma}\psi\\!>$.

If we now take for $\rho$ an invariant density matrix (which in this paper
will be the normalized identity matrix $I$), we define the decoherence matrix
$D$ with entries $D_{\theta,\sigma}(n)$

$D_{\theta,\sigma}(n)=\mbox{Tr}(P^{\dagger}_{\theta}\rho
P_{\sigma})=\mbox{Tr}(P^{\dagger}_{\theta}P_{\sigma})=\frac{1}{\cal
N}\sum_{j=1}^{\cal N}<\\!\psi^{j}_{\theta}|\psi^{j}_{\sigma}\\!>,$ (10)

where ${\cal N}$ is the dimension of the Hilbert space and $\psi^{j}\\!>$ is
an orthonormal basis of ${\cal H}$. In this paper I shall consider a notable
example of such finite-dimensional Hilbert space. The decoherence matrix is a
fundamental object of the coherent histories formalism [44, 45, 46, 47, 48,
49]. Observe that $D(n)$ is an $L^{n}\times L^{n}$ square matrix, Hermitean,
of unit-trace and non-negative. In the classical case, this matrix is
diagonal. In the quantum case, one defines the partition entropy as

$S({\cal H},U,{\cal P},n)=-Tr(D(n)\log D(n))$ (11)

and can therefore proceed to define the AF entropy operating as in equations
(4), (5) [67]. To suit our purposes we do not take the infinite $n$ limit, or
the supremum over partitions, but rather consider the scaling behavior of the
quantity $S({\cal H},U,{\cal P},n)$, concisely denoted $S(n)$ that we will
call AF entropy for short (Shannon AF entropy at finite resolution and time
being a more precise term) for physically accessible partitions, ranges of
time, and physical parameters.

##  4 Dynamical entropy and quantum chaos

On the same theoretical basis as in classical dynamics, we shall consider as
chaotic a quantum system capable of producing new information at an inferiorly
bounded rate indefinitely in time when measured by a sufficiently refined
partition $\cal P$. For this, we adopt the AF quantum entropy defined in the
previous section. It helps to remark that Benatti has derived a quantum
version of the classical Brudno theorem, linking AF entropy and algorithmic
complexity [50], so that the parallelism with classical dynamics is completed.

At a superficial inspection it may seem that there should be no limitation for
the entropy $S({\cal H},U,{\cal P},n)$ in eq. (11) to grow linearly with $n$:
as mentioned, the matrix $D$ has size $L^{n}\times L^{n}$. Yet, thanks to a
result in [26] the spectrum of non-zero eigenvalues of $D$ coincides with that
of a second matrix, $\Omega(n)$, of size ${\cal N}^{2}\times{\cal N}^{2}$
independent of $n$, of unit trace and Hermitean. This matrix is the basis of a
technique to numerically compute $Tr[D(n)\log D(n)]$ combining the theory of
orthogonal polynomials with a sampling method from lattice gauge theory [51].
It appears to be the only method proposed so far capable of attacking
non–trivial Hilbert space and word lengths dimension. While this technique is
rather complicated, one consequence of its is immediate: $S({\cal H},U,{\cal
P},n)$ is the Shannon entropy of a probability distribution over ${\cal
N}^{2}$ discrete states, whose number is independent of $n$. Therefore, it is
bounded by $2\log{\cal N}$ and the limit in eq. (4) is forcefully null.

Why this is a serious problem can be easily understood: consider a classical
dynamical system that can be quantized via the standard procedure of replacing
classical variables by quantum operators. Further assume that the resulting
Hilbert space has finite dimension ${\cal N}$. Consider a classical partition
${\cal P}$ that corresponds to a quantum family of projectors. One can compute
the classical probabilities $\mu({\cal P}_{\mathbf{\sigma}})$ as well as the
eigenvalues of the quantum decoherence matrix $D(n)$ and then compare the
associated Shannon entropies at variable duration $n$ of the experiment. The
correspondence principle leads us to expect that the two results be
comparable, in the classical limit. Yet, recall that the quantum system is
endowed with a finite information reservoir. In the initial instants of the
evolution the classical system is outputting information and the quantum
analogue follows suit, keeping pace with the classical. Yet, this can last as
long as the quantum system has not fully exhausted its information reservoir.

Suppose now that the classical system is not chaotic, in the sense that
$S({\cal P},n)$ grows less than linearly, say logarithmically, with $n$. In
this case, the value of $n$ at which the quantum dynamics can provide the same
information output as the classical is roughly proportional to $\cal N$, and
therefore sufficiently large to justify a physical relevance of
correspondence. On the contrary, when the classical dynamics is chaotic, i.e.
$S({\cal P},n)\sim h({\cal P})\;n$, the finite quantum information reservoir
is eaten away at a rate sustainable only up to $n\sim\log{\cal N}$, after
which the two evolutions qualitatively and quantitatively differ. This is in
essence the logarithmic catastrophe in the correspondence principle mentioned
before.

From a physical point of view, the last estimate corresponds to what has been
defined the Zaslavsky time scale [52] (also termed Ehrenfest time scale [53])
$\tau=\log({\cal I}/\hbar)$, where ${\cal I}$ is a typical action of the
system, a fact that yields the analogy ${\cal N}\sim({\cal I}/\hbar)$ and
prompts matter for thought. In this paper, $\hbar$ will be constant, while the
mass of a quantum particle will determine ${\cal N}$.

From a theoretical point of view, advocated by Berry among others [7, 8], it
is clear that dynamical entropies are properly defined only in the limit when
$n$ tends to infinity, while the classical limit is ${\cal
I}/\hbar\rightarrow\infty$ and the two limits do not commute. At the same
time, it is apparent that this attitude ignores the unphysically low value of
the Zaslavsky time for e.g. the chaotic three body problem. Moreover, as I
have first suggested in [12], rather than considering the limit for $n$
tending to infinity, one should study the scaling relations with respect to
the physical parameters of the information–theoretical quantities (4) and (11)
in realistic ranges of the physical variables. In other words: will a sequence
of physically realizable experiments at finite precision be able to reveal a
significant difference among the two dynamics in the semiclassical region? A
model of this situation is found in the dynamics of classical billiards [54]:
a difference emerges clearly in the behavior of the finite time algorithmic
complexity of symbolic trajectories, when polygonal tables approach chaotic or
integrable curved billiards.

In the next section I briefly recap the details of a much investigated system,
the quantum Arnol’d cat, where this situation emerges with clarity. This
system is also particularly useful to study the arguably most significant way
to explain the emergence of classical properties (among which chaos) in
quantum dynamics: decoherence. In this approach, a quantum system is not
isolated, but coupled to an environment that is responsible for the decay of
the off-diagonal matrix elements of $D$. Yet, in the usual application of this
approach the environment is mathematically represented by a random source, or
an infinite bath of oscillators: in any case a system possessing an infinite
amount of information, and not, to the best of my knowledge, by a quantum
system with finite dimensional Hilbert space. In previous works [41] I have
proposed a scheme by which the environment is composed of the tensor product
of simple low–dimensional systems, so that the dimension of the full, purely
quantum, isolated pair system–environment grows exponentially in the number of
the elements of the latter and can therefore provide a physically reasonable
solution to the logarithmic catastrophe described above. The multi-particle
Arnol’d cat introduced in this paper is a simple model to implement this
scheme. Due to technical difficulties, both theoretical and numerical, much is
still to be understood on the information flow in this system. But first, let
us review the original quantum Arnol’d cat.

##  5 Short review of the quantum Arnol’d cat

Despite the fact that this quantum system has now become classical (pun), it
is interesting to present a derivation that opens the way to its decoherent
generalization. We study a point particle of mass $M$ free to rotate on a
ring, without gravity. This ring is a one–dimensional torus of length $L$ and
position on it is $X\in[0,L)$. Most of the time, the particle evolves freely
at constant velocity. At discrete times, it is subject to a periodic impulsive
force, of period $T$, the relative Hamiltonian being:

$H_{cat}(X,Y,t)=\frac{Y^{2}}{2M}-\alpha\frac{X^{2}}{2}\sum_{j=-\infty}^{\infty}\delta(t/T-j).$
(12)

$Y$ is the conjugate momentum to $X$ and $\alpha$ is a coupling constant.
According to Hamilton’s equations, the effect of the force is to
instantaneously change the particle’s momentum.

Let us show that $H_{cat}$ is the Hamiltonian that yields the Arnol’d cat
mapping. Consider the classical period evolution of the dynamical variables
observed at instants immediately following the action of the impulsive force:

$\left\\{\begin{array}[]{l}X\rightarrow X+\frac{T}{M}Y\\\ Y\rightarrow
Y(1+\frac{\alpha T}{M})+\alpha TX.\end{array}\right.$ (13)

Introduce the rescaled momentum $\tilde{Y}:=\frac{T}{M}Y$, which has the
dimensions of a length. The period evolution operator in the new variables is:

$\left\\{\begin{array}[]{l}X\rightarrow X+\tilde{Y}\\\
\tilde{Y}\rightarrow\frac{\alpha T^{2}}{M}X+(1+\frac{\alpha
T^{2}}{M})\tilde{Y}.\end{array}\right.$ (14)

Require that the rescaled momentum variable $\tilde{Y}$ also be periodic, of
the same period $L$ as of $X$. Choosing $L=1$ and $\frac{\alpha T^{2}}{M}$ to
be an integer fulfills this requirement. Finally, the Arnol’d cat mapping on
the unit two–dimensional torus is obtained by further imposing that

$\frac{\alpha T^{2}}{M}=1.$ (15)

A rigorous mathematical framework for the quantum dynamics of this map can be
found in [56]. I now follow the treatment in [11], with results in agreement
with the original work of Schwinger [55] on the quantum dynamics on a torus.
Because of the $X$ periodicity the wave–functions takes the form

$\psi(X)=\sum_{k}c_{k}\phi_{k}(X)=\sum_{k}c_{k}e^{i2\pi kX/L}.$ (16)

In the above, $\phi_{k}(X):=e^{i2\pi kX/L}$ are momentum eigenfunctions and
$c_{k}$, with $k$ integer, are the expansion coefficients. Next, we consider
periodicity in $Y$ with period $ML/T$. The momentum operator is
$\mathbf{Y}=-i\hbar\partial_{X}$, so that
$\mathbf{Y}\phi_{k}(X)=\frac{kh}{L}\phi_{k}(X)$, $h$ is Planck’s constant and
therefore periodicity in momentum implies that

$c_{k}=c_{k+{\cal N}}$ (17)

for any $k$, where ${\cal N}$ is an integer number:

$\frac{ML^{2}}{T}={\cal N}h.$ (18)

One parameter can be freely chosen and we set $L=1$, with no loss of
generality. One can explain the rationale behind eq. (18): to embed quantum
dynamics in a two dimensional torus of unit periodicity in the $X$ direction,
the periodicity in $Y$ must be an integer multiple of the Planck constant.
Finally, using $c_{k}$ = $c_{k+{\cal N}}$ in eq. (16) implies that $\psi(X)$
takes the form of a periodic train of delta functions at the points $X_{j}$:

$X_{j}=\frac{j}{\cal{\cal N}}+s,\;j=0,\ldots,{\cal{\cal N}}-1$ (19)

where $s$ is an arbitrary constant shift. We shall use the notation $|j>$ for
the delta at such positions.

The Hilbert space of the system is so isomorphic to $\mathbf{C}^{\cal{\cal
N}}$. In the position representation wave functions can be written as

$\psi(X_{j})=\frac{1}{\sqrt{{\cal N}}}\sum_{k=0}^{{\cal
N}-1}c_{k}\phi_{k}(X_{j})=\frac{1}{\sqrt{{\cal N}}}\sum_{k=0}^{{\cal
N}-1}c_{k}e^{-i2\pi kj/{\cal N}}$ (20)

and the discrete Fourier transform maps the above to the momentum
representation:

$c_{k}=\hat{\psi}(Y_{k})=\frac{1}{\sqrt{{\cal N}}}\sum_{j=0}^{{\cal
N}-1}\psi(X_{j})e^{i2\pi kj/{\cal N}},$ (21)

where $Y_{k}=kh$.

An important remark at this point is the following: via eq. (18) the dimension
of the Hilbert space is directly proportional to the mass $M$ of the particle.
Therefore, we can take the classical limit in its correct physical form, by
keeping $\hbar$ to its real physical value and by considering a particle of
larger and larger mass $M$.

Quantum motion induced by the Hamiltonian eq. (12) becomes a unitary operator
in ${\cal H}=\mathbf{C}^{\cal N}$ that has been computed in [11]. In the
position representation, where $\psi$ is the vector $\psi(X_{j})$,
$j=0,\ldots,{\cal N}$, the unitary evolution $U^{0}$ induced by the free
rotation $\frac{Y^{2}}{2M}$ has matrix elements

$U^{0}_{kl}=\frac{1}{\sqrt{{\cal N}}}e^{-(\pi il^{2}/{\cal N})}e^{2\pi
ikl/{\cal N}}.$ (22)

The instantaneous change in momentum induced by the kick is rendered by the
operator $K$ with matrix elements:

$K_{kl}=\frac{1}{\sqrt{{\cal N}}}e^{i\pi l^{2}/{\cal N}}\delta_{k,l},$ (23)

where $\delta_{k,l}$ is the Kronecker delta. To complete the review of the
formulae in [11], the quantum Arnol’d cat evolution operator is the product
$U=KU^{0}$.

##  6 The Alicki-Fannes entropy of the quantum Arnol’d cat

The theory of the Alicki-Fannes entropy can be applied to the dynamics of the
quantum Arnol’d cat, using projections operators $P_{l}$ measuring the
position coordinate only: letting $|j>$ for $j=0,\cdots,{\cal N}-1$ be the
canonical basis vectors of $\mathbf{C}^{\cal{\cal N}}$ and letting for
convenience ${\cal N}=2^{q}$, $Q=2^{q-p}$ with $p$ and $q$ positive integers,
$p<q$, we define

$P_{l}=\sum_{j=lQ}^{(1+l)Q-1}|j><j|,\;\;\;l=0,\ldots,2^{p}-1.$ (24)

For illustration, $P_{0}$ is the projection on the first $Q=2^{q-p}$ basis
states, which corresponds to the classical projection on the phase-space set
$0\leq X<1/2^{p}$, that is, the leftmost $1/2^{p}$ section of the unit two-
dimensional torus (notice that all values of $Y$ are allowed).

Even though the adaptation of the formalism of Section 3 to the case of the
quantum Arnol’d cat is theoretically straightforward and physically
transparent, its practical implementation, in terms of numerical analysis, is
particularly challenging. Details of the sophisticated technique required for
its implementation can be found in [51]. Following the approach presented in
[26], this technique permits to compute the entropy for arbitrarily long times
avoiding the exponential increase of the dimension of the decoherence matrix.
In Figure 3 the quantity $S(n)$ (shorthand notation for $S({\cal H},U,{\cal
P},n)$) is plotted versus $n$ for different values of $q$ (recall, ${\cal
N}=2^{q}$) and $p=2$. We see that the bound $S(n)\leq 2\log({\cal N})$ is
approached at a rapid pace in $n$ for any finite value of $q$. Before this
saturation, which happens when $n$ is proportional to $q$, that is $\log({\cal
N})$, there is a small interval when $S(n)$ increases linearly with a slope
dictated by the classical Lyapunov exponent. After this brief lapse of time,
output of new information fades.

Figure 3: AF entropy $S(n)$ as a function of $n$ and $q$.

This is the logarithmic catastrophe described above. For indeed to linearly
increase the length of this span of chaotic freedom one has to exponentially
increase the dimension of phase space, i.e. the mass of the particle! It is
clear that, while mathematically the classical complexity is fully regained in
the limit, this is quite unsatisfactory from a physical viewpoint.

A detailed analysis of the decoherence matrix yielding these results further
reveals their physical content. In the numerical experiments that follow, we
fix $p=2$, meaning that phase space and Hilbert space are partitioned in four
sets (see eq. (24) and discussion) and consequently the decoherence matrix is
the direct sum of sixteen diagonal blocks, of which we consider the block
$\sigma_{0}=\theta_{0}=0$, $\sigma_{n-1}=\theta_{n-1}=0$, the other blocks
providing similar results. In Figure 4, left panel, the matrix elements
$|D_{\sigma,\theta}|$ are plotted as blue bars. The wordlength is three, so
that interference between the paths $(0,\sigma_{1},0)$ and $(0,\theta_{1},0)$,
as in a four-slit experiment, is clearly evident. Because of this, the
diagonal values compare poorly with the classical probabilities $\mu(\sigma)$
(a shorthand notation for $\mu(P_{\sigma})$), plotted as red bars on the
diagonal. The forest of off-diagonal blue bars is even more evident when
considering words of length four $(0,\sigma_{1},\sigma_{2},0)$ (there are 16
of them), in the right panel of Figure 4.

|
---|---
Figure 4: Absolute value of the decoherence matrix elements $D_{\sigma,\theta}$ (blue bars) and classical word probabilities $\mu(\sigma)$ (red bars, plotted on the diagonal for comparison), when $q=2$ so that ${\cal N}=2^{q}=4$, and $p=2$. In the left panel $|\sigma|=|\theta|=3$, in the right panel $|\sigma|=|\theta|=4$. See text for further detail. |
---|---
Figure 5: Absolute value of the diagonal decoherence matrix elements
$D_{\sigma,\sigma}$ (blue bars) as function of $q$, compared with the
classical word probabilities $\mu(\sigma)$ (red bars, obviously independent of
$q$ and plotted sideways for comparison). In the left panel $|\sigma|=3$, in
the right panel $|\sigma|=4$. See text for further detail.

How does then the classical limit arise in this context? In Figure 5 the
diagonal matrix elements $D_{\sigma,\sigma}$ are plotted versus $q$, and
compared with the classical probabilities $\mu(\sigma)$. Since the mass $M$ is
proportional to ${\cal N}=2^{q}$, increasing $q$ by one unit means doubling
the mass of the particle. Convergence to the classical values is observed as
well as the progressive quenching of off-diagonal matrix elements. In fact, in
Figure 6, left panel, it is seen that when $q$ increases the decay of the sum
of the absolute values of the off-diagonal matrix elements
$D_{\sigma,\theta}$, $\sigma\neq\theta$, goes in pair with convergence to the
classical entropy $S_{cl}$ of the diagonal entropy
$S_{diag}=-\sum_{\sigma}D_{\sigma,\sigma}\log D_{\sigma,\sigma}$ and of the AF
entropy $S_{AF}$ in eq. (11). The fitting line in the figure reveals that
these phenomena scale as a power law with the mass $M$ (proportional to
$2^{q}$), with exponent between minus one and minus one half.

The right panel of Figure 6 reveals in even clearer detail the logarithmic
catastrophe alluded before. The symbolic distance, $d_{\mbox{\tiny
symb}}=\sum_{\sigma}|D_{\sigma,\sigma}-\mu(\sigma)|$, at fixed word length
$|\sigma|$, is plotted when both $|\sigma|$ and $q$ are varied. One observes
that classical word probabilities of fixed length $|\sigma|$ are recovered
when increasing the mass of the particle (i.e. $q$), while quantum
interference destroys this convergence when, at fixed mass, longer words are
considered. This is the well-known non-commutativity of the classical limit
and the long-time limit [7, 8]. Yet, the graph shows that quantum effects
spoil the correspondence at a word length that is logarithmic in the mass of
the particle. One so recovers from the entropic point of view the results
obtained via algorithmic complexity [11].

|
---|---
Figure 6: Left panel: for $|\sigma|=3$, sum of the absolute values of the off-
diagonal matrix elements $D_{\sigma,\theta}$ (blue curve), absolute difference
between classical entropy $S_{cl}$ and diagonal entropy $S_{diag}$ (red
curves), absolute difference between classical entropy $S_{cl}$ and AF entropy
$S_{AF}$ (green curves). Right panel: symbolic distance $d_{\mbox{\tiny
symb}}=\sum_{\sigma}|D_{\sigma,\sigma}-\mu(\sigma)|$ versus word length
$|\sigma|$ and $q$. See text for further detail.

As mentioned, decoherence has been proposed as a way to overcome this impasse.
It is the purpose of the remaining part of this paper to study decoherence in
a controlled setting, with the aid of a model system.

##  7 Decoherence and the multiparticle Arnol’d Cat

Decoherence, that is, the loss of quantum coherence of a system, is today
principally studied because it is a major obstacle in the performance of a
quantum computer [57]. In the nineties of the XX century was invoked for the
explanation of the puzzling behavior of quantum Schrödinger cats [58] and the
emergence of classical properties in quantum mechanics [42], [17, 18, 19, 20,
21]. Actually, as early as 1984 Guarneri [59] proved that a random
perturbation of the motion of a kicked rotor prevents quantum localization to
set in. This induces a diffusive quantum motion, whose diffusion coefficient
$D$ was computed by Ott et al. [60]. In the same perspective, Dittrich and
Graham [61], Kolovsky [62] and Sundaram et al. [65] considered systems (both
classical and quantum) randomly perturbed or coupled to an environment: the
resulting motion is also diffusive. The quantum dynamics of these system was
shown to be similar to the classical when $\sqrt{D/\lambda}$ is larger than
$\hbar$ times a dimensional constant ($\lambda$ is the Lyapunov exponent). The
heuristic explanation of this inequality stems from the observation that since
diffusion smears classical phase space to the finest discernable resolution of
$\sqrt{D/\lambda}$, when this latter is larger than $\hbar$, the quantum
smeariness, the two dynamics appear similar. To quote a famous musical, The
way she sees and the way he looks is a perfect match: the reader is asked to
interpret the analogy. We shall return to this point in the conclusions.

Let us now describe a system that I have introduced to model exactly the
interaction of a system with the environment, avoiding recourse to master
equations, Lindblad operators, or the like, with the goal of having a fully
Hamiltonian system [41]. The model draws upon the original view of decoherence
by Joos and Zeh [42, 68] as due to collisions of particles.

Consider adding $I$ smaller particles of mass $m$ in the ring that confines
the particle of mass $M$, freely rotating except for collisions with the large
particle. Let $x_{i}$ and $y_{i}$, $i=1,\ldots,I$ the coordinates and momenta
of these particles, respectively. The Hamiltonian of the classical system is
therefore

$H=H_{cat}(X,Y,t)+\sum_{i=1}^{I}\frac{y_{i}^{2}}{2m}+\kappa\sum_{i=1}^{I}V(x_{i}-X),$
(25)

where $H_{cat}$ has been introduced in eq. (12), $\kappa$ is a coupling
constant and the potential function $V(\cdot)$ will be described momentarily.

Let us first define the quantum kinematics. In addition to the $L=1$
periodicity in the variables $x_{i}$, we also impose periodicity
$\frac{mL}{T}$ in the momenta $y_{i}$. Therefore, performing a similar
analysis with the one presented above, we obtain

$\frac{mL^{2}}{T}={\nu}h,$ (26)

where $\nu$ is the dimension of the Hilbert space of a single small particle,
an integer that we also choose to be a power of two: $\nu=2^{r}$. Therefore,
the manyparticle wave–function reads

$\Psi(X,x_{1},\ldots,x_{I})={\cal N}^{-1/2}\nu^{-I/2}\sum_{k_{0}=0}^{{\cal
N}-1}\sum_{k_{1}=0}^{\nu-1}\cdots\sum_{k_{I}=0}^{\nu-1}c_{k_{0},k_{1},\ldots,k_{I}}e^{2\pi
i(k_{0}X+\sum_{i}k_{i}x_{i})}.$ (27)

In the same way as the position $X$ is restricted to a lattice, so are the
$x_{i}$’s. We further require that small particles’ positions $x_{i}$,
$i=1,\ldots,I$, be a subset of those of the large one:

$x_{i}^{j_{i}}=j_{i}\frac{1}{\nu}+s_{i}\frac{1}{{\cal
N}},\;\;j_{i}=0,\ldots,\nu-1.$ (28)

where $s_{i}$ an integer ranging from zero to ${\cal N}/\nu-1$. Introducing
the multi-index ${j}={j_{0},j_{1},\ldots,j_{I}}$, in which the label $0$
refers to the large particle and $i>0$ to the $i$-th small one, we let $|{j}>$
denote the delta function at the particles’ position given in equations (19),
(28). A different representation for the wave function is therefore

$\Psi=\sum_{j_{0}=0}^{{\cal
N}-1}\sum_{j_{1}=0}^{\nu-1}\cdots\sum_{j_{I}=0}^{\nu-1}d_{j_{0},j_{1},\ldots,j_{I}}|j_{0},j_{1},\ldots,j_{I}>.$
(29)

The Hilbert space of the system ${\cal H}$ is the tensor product of the
Hilbert spaces of the single particles, and therefore it is isometric to a
power of $\mathbf{C}$:

${\cal H}={\cal H}_{big}\otimes{\cal H}_{small}^{I}={\mathbf{C}}^{{\cal
N}I^{\nu}}.$ (30)

The potential $V$ must represent a short–range interaction occurring only when
the large particle and a small one occupy the same position. In the
representation eq. (29) its matrix elements are therefore

$<{j^{\prime}}|V{j}>=V_{j^{\prime}_{0},j^{\prime}_{1},\ldots,j^{\prime}_{I};j_{0},j_{1},\ldots,j_{I}}=\prod_{i=0}^{I}\delta_{j^{\prime}_{i},j_{i}}\sum_{i=1}^{I}\delta_{j_{0},\frac{\cal
N}{\nu}j_{i}+s_{i}}.$ (31)

##  8 The Von Neumann Entropy

In a pair of papers I have investigated the dynamics of the quantum system
introduced in the previous section [41, 43]. Treating the large, zeroth
particle as the system and the small particles as the environment, the reduced
density matrix $\hat{\rho}$ is defined by tracing over the small particles
degrees of freedom:

$\hat{\rho}_{j^{\prime}_{0},j_{0}}=\sum_{j_{1},\ldots,j_{I}}\rho_{j^{\prime}_{0},j_{1},\ldots,j_{I};j_{0},j_{1},\ldots,j_{I}}$
(32)

Firstly, by choosing an initial density matrix as a pure state in a
Schrödinger cat configuration [58], I showed that as time progresses,
interaction with small particles wipes out the outdiagonal components of the
reduced density matrix $\hat{\rho}$, obtained by tracing over the small
particles degrees of freedom, exactly as in the experimental results. This
dynamics is shown in the position representation in Figure 7. Here,
$\alpha=0$, that is, there is no cat kick. These results show that the
multiparticle cat is a bona fide model of decoherence.

|  |
---|---|---
|  |
Figure 7: Absolute value of the reduced density matrix, in the position
representation, of a system with $q=7$, $I=2$, $\nu=1$, $\kappa=50$. Time
snapshot ordered from left to right, top to bottom. The cat particle has a
positive velocity, so that peaks drift on the torus.

I then considered the Von Neumann entropy,
$-\mbox{Tr}\;\hat{\rho}\log\hat{\rho}$. This entropy quantifies the amount of
information in the eigenvalues of the reduced density matrix. The analysis was
first performed for the free colliding particles case, given by the
Hamiltonian (13), (25) with $\alpha=0$. The initial increase of the Von
Neumann entropy was observed to be logarithmic. On the contrary, when $\alpha
T^{2}/M=1$, as in the Arnol’d cat map, the initial increase of entropy is
linear, with a slope that equals the Kolmogorov Sinai entropy provided the
coupling constant $\kappa$ (Eq. 25) takes a specific value, which scales
according to the physical parameters of the system. Clearly, when $\kappa$ is
null there is no decoherence and the Von Neumann entropy is null. For small
values of $\kappa$ entropy increases linearly at a small rate before
saturation, while above the aforementioned value the rate is larger than the
classical dynamical entropy. This parameter dependence is in my view a caveat
for the physical interpretation of similar results in different systems. For
details, the interested reader is referred to [43].

##  9 The AF Entropy of the multiparticle Arnol’d cat

Even though interesting, results about the Von Neumann entropy lack the
generality and theoretical significance provided by the Alicki-Fannes entropy.
In particular, the former is bounded above by the logarithm of the dimension
of the Hilbert space of the reduced system, in this case $\log{\cal N}$,
independently of the dimension of the full Hilbert space. The latter, on the
contrary is bound by $2\log{\cal N}+2I\log(\nu)$, which scales linearly with
the number of particles $I$ and therefore might provide a physically
reasonable classical limit.

Moreover, the decoherence matrix introduced in Sect. 3 permits to compare
analytically, i.e. one by one the classical probabilities $\mu(P_{\sigma})$
with the diagonal elements $D_{\sigma,\sigma}$ and to gauge the importance of
non diagonal matrix elements, which encode quantum coherence. Observe that the
projectors $P_{l}$ in eq. (24) can be generalized to act on the coordinate of
the large particle only, labeled by the index zero:

$P_{l}=\sum_{j_{0}=lQ}^{(1+l)Q-1}\sum_{j_{1}=0}^{\nu-1}\cdots\sum_{j_{I}=0}^{\nu-1}|j_{0},j_{1},\ldots,j_{I}><j_{I},\ldots,j_{1},j_{0}|,\;\;\;l=0,\ldots,2^{p}-1.$
(33)

In so doing, we are observing the symbolic dynamics of the large particle
only, in the presence of the external environment provided by the other,
smaller, particles.

###  9.1 The decohering effect of the perturbation

|
---|---
Figure 8: Absolute value of the decoherence matrix elements
$D_{\sigma,\theta}$ (blue bars) and classical word probabilities $\mu(\sigma)$
(red bars, plotted on the diagonal for comparison) when $q=4$, $r=1$, $I=2$
and $p=2$. In the left and right panels $\kappa=0$ and $\kappa=10$,
respectively. See text for further detail.

Firstly, let us consider a system composed of a large particle, $q=4$, coupled
to two small particles, $r=1$ (mass ratio 1:8) and let us vary the coupling
constant $\kappa$. In Figure 8 we plot as before the decoherence matrix
entries for words of length 5, beginning and ending with the symbol 0. In the
left panel $\kappa=0$ (that is, small particles do not interact with the large
one), while in the right panel $\kappa=10$. Non diagonal matrix elements are
clearly smaller in the second case, so that diagonal values become closer to
the eigenvalues of the decoherence matrix. Nonetheless, we do not expect them
to approach in the limit of very large $\kappa$ the classical word
probabilities, displayed as red bars in both panels. This is evident in Figure
9 where the diagonal matrix elements $D_{\sigma,\sigma}$ are plotted versus
$\kappa$ and compared to $\mu(\sigma)$. The symbolic distance $d_{\mbox{\tiny
symb}}=\sum_{\sigma}|D_{\sigma,\sigma}-\mu(\sigma)|$ quantifies the agreement:
in the right panel of Figure 9 it is seen that $d_{\mbox{\tiny symb}}$ does
not significantly diminishes when increasing $\kappa$, differently from the
sum of the absolute values of the off-diagonal matrix elements,
$\sum_{\sigma\neq\theta}|D_{\sigma,\theta}|$.

I believe that these results are general, albeit obtained in a restricted
range of parameters. Firstly, increasing $\kappa$ reduces the off-diagonal
matrix elements and therefore makes the system less coherent, i.e. more
classical. Secondly, observe that the sums $\sum_{\sigma}D_{\sigma,\sigma}$
and $\sum_{\sigma}\mu(\sigma)$ over words of fixed length $|\sigma|$ are
constant and equal to the probabilities, quantum and classical, of the sector
$\sigma_{0},\sigma_{n-1}$, ($0,0$ in the figures) so that changing the value
of $\kappa$ results in a transfer of probability among all words of length
$n=|\sigma|$ while keeping their sum approximately constant (because transfer
of probability among different blocks $\sigma_{0},\sigma_{n-1}$ is small).
Therefore, the fact that increasing $\kappa$ at fixed $|\sigma|$ does not
reduce the symbolic distance means that small particles are rendering the
observation of the large one more classical, but at the same time are
perturbing it too much. Finally, the increase of the symbolic distance with
$|\sigma|$ (an evident proxy for time) at fixed $\kappa$ shows again the
rapidly emerging difference between classical and quantum dynamics. Figure 10
confirms this conclusion: we see that in the presence of decoherence (that is,
low off-diagonal matrix elements) AF and diagonal entropies approach each
other, but differ from the classical value.

|
---|---
Figure 9: Left panel: Parameters as in Fig. 8. Diagonal matrix elements $D_{\sigma,\sigma}$ (red bars) versus $\kappa$, classical word probabilities $\mu(\sigma)$ (green bars), and sum of the absolute value of outdiagonal matrix elements $D_{\sigma,\theta}$ (blue bars, rescaled by a factor to improve readability of the graph). Right panel: sum of the absolute differences between diagonal matrix elements and classical probabilities (red), sum of the absolute values of off-diagonal matrix elements (green, rescaled by a factor). |
---|---
Figure 10: Parameters as in Fig. 8, wider range of the coupling constant
$\kappa\in[0,40]$. Left panel: classical entropy $S_{cl}$ (red curves),
diagonal entropy $S_{diag}$ (green curves), AF entropy $S_{AF}$(blue curves).
Right panel: absolute difference $|S_{cl}-S_{diag}|$ (red curves) and
$|S_{cl}-S_{AF}|$ (green curves).

###  9.2 The limit of large mass with decoherence

The previous results have shown that there exist two ways to induce
decoherence, defined as quenching of the off-diagonal entries of the matrix
$D$: increasing the mass of the particle (that is, the dimension of its
Hilbert space), or coupling it to a larger, yet still purely quantum system.
The former leads in the limit to the classical results, but is plagued by its
very slow, logarithmic convergence, which renders its application unphysical.
The second cannot provide by itself the hardly-sought classical limit, but it
may feature an exponential increase in Hilbert space dimension, simply
augmenting the number of small particles. I made this observation in [41],
where I surmised that a combination of the two approaches might yield scaling
relations (in the number of small particles, coupling, and mass of the large
particle) by which complexity of the motion, as a linear increase of the
Shannon Alicki-Fannes entropy $S(n)$, could be observed over physically
significant time spans.

This investigation is subtle, both from the theoretical and the numerical
viewpoint. In this paper we can add a few experiments in this direction. In
Figure 11 we consider two families of systems, the former composed of a single
particle, of increasing mass (different values of $q$), already studied in
Section 6; the second, composed again of such large particle, but coupled to
two small particles, with coupling constant $\kappa$ that increases
proportionally to the mass of the large one. We compute the AF entropies
$S_{AF}$ of these systems and compare them with the classical entropy. We
observe that, while both converge to the last quantity, the multiparticle case
provides a better approximation for small values of $q$. Considering that the
mass $M$ is proportional to the exponential of $q$, this is encouraging in the
quest outlined above.

Figure 11: Classical entropy $S_{cl}$ (red curves) compared with AF entropy $S_{AF}$ of systems with variable $q$ and $I=0$ (green curves) and $I=2$, $r=1$ and $\kappa=.125\times 2^{q}$ (black curves).  |
---|---
|
|
Figure 12: Increasing partial entropies $S_{i}$, $|\sigma|=5$, of systems with
variable $q$, from 2 to 7, left to right and top to bottom. The classical
entropy $S_{i}$ is denoted by a red curve in all panels. In the case $I=0$ the
green curve marks the AF partial entropy, and the blue curve the diagonal
partial entropy. In the case $I=2$, $r=1$ and $\kappa=.125\times 2^{q}$ the AF
partial entropy is marked in magenta and the diagonal partial entropy in light
blue.

To investigate this phenomenon, recall that, as described above, for any
finite value of $n=|\sigma|$ and for any finite sector
$\sigma_{0},\sigma_{n-1}$, ($0,0$ in the figures) the AF, diagonal, and
classical entropies are the sum of $L^{n-1}$ contributions of the form
$s_{j}=-\zeta_{j}\log(\zeta_{j})$, when $\zeta_{j}$ is an eigenvalue of the
matrix $D$, or a diagonal entry of this matrix, or a classical word
probability, respectively. We can therefore compare different entropies by
first sorting the values $s_{j}$ in descending order, from the largest to the
smallest, and then computing their integrated distribution
$S_{i}=\sum_{j=0}^{i}s_{j}$. These distributions are plotted in Figure 12, in
the same case of Figure 11. Different panels are related to increasing values
of $q$, from 2 to 7. The horizontal segments in the plots of the AF partial
entropies are due to the Hilbert space dimensionality bound, which imposes a
maximum number of non-null eigenvalues of $D$. We can observe that the effect
of the small particles is to free the AF entropy from this bound, assuring a
faster convergence to the classical values.

##  10 Conclusions

The multiparticle Arnol’d cat is a model system particularly suited to
investigate the nature, definition and relevance of quantum chaos. In my works
I have followed the original suggestion of Chirikov and Ford that this problem
should be considered within the framework of information theory. Dynamical
entropies provide a major tool for this investigation and their formal aspects
have been fully developed in the last years [67]. In this respect, the model
described herein permits to free dynamics in a controlled, fully canonical
way, from the bound induced by the finite dimensionality of the Hilbert space.
This bound is due, in this particular model, by the finiteness of the mass of
a particle, and therefore carries a deep physical significance.

In fact, I considered herein the classical limit as the limit of increasing
mass of such particle, while keeping the value of the Planck constant fixed.
Chaotic behavior has been defined, for a finite time span, as the production
of information at an inferiorly bounded, positive rate. Consideration of the
decoherence matrix from the consistent histories formalism has also permitted
to compare word by word classical and quantum symbolic dynamics. It came as no
surprise that, as I presented in earlier works, in the single particle case
the length of agreement between the two, and consequently the length of the
span of the chaotic freedom of the quantum system, grows only logarithmically
with the mass of the particle.

As reviewed in this paper, coupling to an external system has been shown to
provide quantum motions (via the Wigner function, or similar phase space
tools) that correspond to a classical noisy evolution, within the finite
phase-space resolution of this latter, for longer times. In a discrete lattice
model, mimicking the quantum cat Hamiltonian with noise [69] we have observed
information production over long time scales. Returning now to many particles
quantum systems, in [41] I put forward the hypothesis that a particular
scaling of parameters: number of particles, their mass, and the observational
time interval, could extend in a physically reasonable range the span of
chaotic behavior. The results of this paper are admittedly not conclusive, but
suggest that this endeavor could be feasible. In this view, the (fully
quantum) environment is to be considered as a source of disorganized
information that, when input to a regular or chaotic quantum system, is
organized by this latter in such a way to agree, within observational limits,
to the classical description. This is in essence the program outlined in [69]:
For a long time, research in quantum chaos has looked for quantum
characteristics related to classical chaotic motion. The fact that none of
these could be properly called chaos led to the concept of pseudochaos and
cast doubts on the very existence of chaos in nature. [..] one might try to
reverse this approach and consider classical dynamics as an effective theory
that, via truly chaotic deterministic dynamical systems, models a randomly
perturbed quantum motion under observational coarse graining.

##  11 Acknowledgements

Partially funded from grant PRIN 2017S35EHN Regular and stochastic behavior in
dynamical systems. Computations for this paper were performed on the Zefiro
cluster at the INFN computer center in Pisa. Enrico Mazzoni is warmly thanked
for assistance.

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# A liquid-phase loop-mode argon purification system

Christoph Vogl,11footnotetext: Corresponding author. Mario Schwarz  Patrick
Krause  Grzegorz Zuzel  and Stefan Schönert

###### Abstract

Noble gas and liquid detectors rely on high chemical purity for successful
operation. While gaseous purification has emerged as a reliable method of
producing high-purity noble fluids, the requirement for large mass flows
drives the development of liquid-phase purification. We constructed a medium-
scale liquid argon (LAr) purification system based on a copper catalyst and 4
molecular sieve capable of purifying 1 of commercial LAr 5.0 to a long
effective triplet lifetime of $\tau_{3}\sim\qty{1.3}{\micro\s}$. We further
demonstrate that a quenched effective triplet lifetime of
$\tau_{3}\sim\qty{1}{\micro\s}$, due to contamination by air, can be recovered
in loop-mode purification to $\tau_{3}\sim\qty{1.3}{\micro\s}$ after > 20
volume exchanges.

##  1 Introduction

Impurities deteriorate the performance of noble gas and liquid detectors.
Nitrogen and oxygen decrease the light yield of scintillation detectors
through quenching processes [1, 2], oxygen and water absorb VUV photons [2,
3], and electronegative impurities capture ionization electrons in time-
projection-chambers [4]. The continuous increase in detector size and the
resulting need to transport VUV photons and ionization electrons over long
distances lead to stringent purity requirements. Purifying noble gases and
liquids has become a standard experimental technique in astroparticle physics.

Most liquid noble gas experiments perform purification in the gas phase with a
commercial purifier based on special getter alloys operated at a high
temperature of around 400 [5]. The gas is either drawn from the ullage or
evaporated from the liquid phase. It is cleaned in a hot getter, condensed in
a heat exchanger, and fed back to the liquid volume. Experiments such as
DarkSide-50 [6] and XENON1T [7] have demonstrated impurity concentrations on
the sub-parts-per-billion level in liquid argon and xenon using this
technique.

Due to limited throughput and growing detector mass, gaseous purification is
reaching its limits, fostering interest in liquid-phase purification where
much larger mass flows are possible (the density of, e.g., argon is $\sim 780$
times higher in the liquid phase at the boiling point than in the gaseous
phase at standard temperature and pressure). Liquid-phase purification is the
method of choice for giant liquid argon time-projection-chambers studying
neutrinos at reactors and particle beams [8, 9, 10, 11, 12, 13]. However, also
low-background experiments as XENONnT [14] and Legend-200 [15, 16] employ
liquid-phase purification.

This paper presents the LAr purification system Lapis (Liquid Argon
Purification Instrument for Scarf222Scarf is the Subterranean Cryogenic ARgon
facility [17], a 1 LAr cryostat used for R&D for Legend and previously for
Gerda.) at the shallow underground laboratory of the Technical University of
Munich. It is a down-scaled sibling of Llars (the Legend Liquid ARgon
purification System), provides the possibility to test and develop
purification strategies on a more accessible scale than in the 90 LAr tank of
Legend-200, and supplies high-purity LAr for germanium detector tests,
scintillation and doping studies.

##  2 Experimental setup

The highest widely available quality of liquid argon is grade 5.0 which
contains up to 2 oxygen and 3 water333Due to the possible confusions between
ppm (part-per-million) by amount and ppm by mass, we chose to denote
concentrations by , identical to ppm by amount., leading to strong quenching
of the argon excimers and an effective triplet lifetime down to
$\tau_{3}<\qty{400}{\nano\s}$. Lapis is designed to purify up to 1 of LAr 5.0
such that the effective triplet lifetime is not quenched anymore. This
corresponds to a value of $\tau_{3}\sim\qty{1.3}{\micro\s}$, and oxygen
contaminations below 0.01 [2]. Purification is possible in batch mode, i.e.,
during the filling of Scarf, or in loop mode using a submerged cryogenic LAr
pump at the bottom of the cryostat.

Lapis contains two traps. One filled with 2.3 Q-5 copper catalyst ($14\times
28$ mesh) by Research Catalysts [18] to remove oxygen, and the other filled
with 2.0 4 molecular sieve (1.6 to 2.6 large beads) from Sigma Aldrich [19] to
remove water. Copper captures oxygen via the reaction O2 + 2Cu -> 2CuO,
whereas water attaches to the molecular sieve through van der Waals forces.
Limited amounts of nitrogen are trapped by these substances as well.

The traps are 350 long stainless steel cylinders, metal-sealed by CF100
flanges. Sintered metal filters with 10 pore size set inside the flanges hold
the purification substances in place, and custom-made dispensers distribute
the LAr flow over the cross-section. Figure 1 shows the piping and
instrumentation diagram of the system.

Figure 1: Piping and instrumentation diagram of Lapis, the Liquid Argon
Purification Instrument for Scarf. LAr is pushed into the system from the
bottom, enters first the water trap, then the oxygen trap and exits at the
top. The equipment for regeneration, i.e., electrical heaters, reduction gas,
and a vacuum pump, are also shown. For details, see text.

The LAr input is located at the bottom valve (V1) and provided either by the
LAr pump (loop mode) or by a LAr tank delivery (batch mode). The water trap is
located upstream because the oxygen trap is more efficient in the absence of
moisture [8]. The LAr flows upwards (i.e., against gravity) through the
purification columns to facilitate a homogeneous distribution over the cross-
section of the traps.

The traps are equipped with electrical heaters and two temperature sensors:
One in the center and one on the heater. Before the first use, and when the
purifier is saturated with impurities, it must be regenerated. The molecular
sieve is cleaned by elevating its temperature to 250 and applying vacuum. The
oxidized copper is heated to 175225 and flushed with hydrogen diluted in an
inert carrier gas. The vendor recommends a concentration of hydrogen of 1.52
[20] to prevent thermal damage during the exothermal reduction reaction,
however often concentrations up to 5 are used. Initially, we used 5 hydrogen
in nitrogen but switched later to 2 hydrogen in argon. Flushing with hydrogen
reduces the copper-oxide via $\ce{2CuO+H2+heat->2Cu+2H2O}$ to pure copper and
water. Monitoring the water content in the exhaust permits to determine when
the regeneration is complete. Around 800 trap volumes must be exchanged to
guarantee a complete reduction of the copper catalyst using a 5 hydrogen
mixture [15]. To reduce heat load from the environment, Lapis is insulated
with rubber foam along the LAr path.

##  3 Batch-mode purification

Lapis was used to fill Scarf with 1 of LAr from a commercial vendor. We
purified two 600 tanks containing LAr 5.0 in batch mode in 34 each. The
initial impurity concentrations were certified by the vendor to be 0.5 (0.3)
oxygen and 0.5 (0.2) nitrogen for the first (second) tank. According to
literature [2, 1], an effective triplet lifetime of
$\tau_{3}<\qty{1.0}{\micro\s}$ is expected. We measure $\tau_{3}$ of the
purified LAr in Scarf with the Legend Liquid Argon Monitoring Apparatus, Llama
[21, 22], an in-situ device based on a silicon-photomultiplier array developed
for and integrated in Legend-200 to monitor the LAr optical parameters. An
identical system is available for Scarf. After purification we find
$\tau_{3}=\qty{1.31(2)}{\micro\s}$, demonstrating the purification
capabilities of Lapis.

##  4 Loop-mode purification

After several months of Scarf operation, an air leak decreased $\tau_{3}$ to
0.98(2), and we decided to use Lapis to restore the scintillation properties
using a cryogenic submersible pump installed at the bottom of the cryostat. It
is a custom-built piston pump by the Institute of Air Handling and
Refrigeration in Dresden, Germany [23], and based on a linear drive. The
volume flow can be regulated between 75 and 495 by defining the frequency of
the piston’s movement. To reduce the amount of boil-off gas, especially during
the cooldown of the purification system, we installed heat-exchanging copper
coils at the traps and cool them down with liquid nitrogen during and before
operation.

LAr is extracted from the bottom of Scarf, pumped through Lapis, and
reinserted above the liquid. Llama is continuously monitoring the optical
parameters. Four loop-mode purification runs were necessary to restore
$\tau_{3}$ to its former value of $\sim\qty{1.3}{\micro\s}$. In total, more
than $20$ volumes were exchanged, vastly more than expected. With perfect
mixing and impurity trapping, a reduction in impurity concentration by
$1/\mathrm{e}$ per volume exchange can be derived. We tentatively assign the
inefficient purification to nitrogen impurities, since the contaminant is
known to be air and Lapis is not optimized for nitrogen removal.

Figure 2: Evolution of the effective triplet lifetime (top) and p.e.-yield
(bottom) measured by Llama during loop-mode purification. The LAr pump runs in
the timeframes between two black vertical lines and is off in the last panel.
The dashed orange line denotes an adjustment of volume flow in panel four. The
x-axis labels show the time since the pump is running. For details, see text.

The first three panels in Figure 2 show the effective triplet lifetime and the
photo-electron yield as measured by Llama in single-day purification runs
where we pre-cooled Lapis in the morning, started the pump, purified six to
eight hours and turned the system off overnight. Several different volume
flows were tested: 450 in the first panel, 300 in the second, and 225 in the
third. While the enhancement in scintillation parameters is most pronounced in
the third panel with the lowest volume flow, the nonlinear dependence on the
impurity concentrations necessitates more careful modeling to determine the
optimal purification speed. Before and after the first run, Lapis was fully
regenerated. The molecular sieve specifically was regenerated another time
after the third run.

Panel four shows a 42-long run where we kept the system on continuously. The
sudden drop of both scintillation parameters between the end of panel three
and the beginning of panel four is currently not understood but might be
caused by an inflow of additional impurities at system startup. The volume
flow is 150 for the first 28.5 (until the dashed orange line) and 225 for the
rest. The effective triplet lifetime recovers its original value of
$\sim\qty{1.3}{\micro\s}$, whereas the p.e.-yield is only 29.21(2)pe at the
end, compared to 34.56(1)pe directly after filling. The origin of this
difference is not clear. Quenching can be excluded due to the long effective
triplet lifetime, and attenuation in the LAr bulk is unlikely given that the
distance between the light source and the light detectors used to extract the
p.e.-yield is only a centimeter in Llama. A possible explanation could be the
presence of a thin, strongly attenuating layer of frozen water on the surface
of the detectors.

The last panel of Figure 2 shows the evolution of the scintillation parameters
after we shut down the pump. $\tau_{3}$ is constant, but the more sensitive
p.e.-yield is decreasing slightly, possibly due to residual outgassing.

Due to the currently poor thermal insulation of Lapis, LAr was evaporating
during the runtime of the pump and ejected into the atmosphere. We estimate to
have lost 365 of LAr. This will be mitigated in the future by moving the
system into a vacuum-insulated cold box.

##  5 Conclusion

We presented Lapis, the Liquid Argon Purification Instrument for Scarf, and
provided two demonstrations of its usage: Batch-mode purification while
filling and loop-mode purification thereafter. Liquid argon contaminated with
air, featuring a quenched effective triplet lifetime of
$\tau_{3}=\qty{0.98(2)}{\micro\s}$ was restored with loop-mode purification
using dispersed copper and molecular sieve 4. $\tau_{3}$ was raised back to
its original value of 1.3. However, > 20 volume exchanges were necessary to
achieve that, vastly exceeding expectations. The experience gathered here
provides valuable insight into applying LAr loop-mode purification for the
Legend collaboration, which owns a similar but larger system.

## Acknowledgments

This work has been supported in part by the German Federal Ministry for
Education and Research (BMBF) within the project 05A20WO1 and the Polish
National Science Centre (Grant No. UMO-2020/37/B/ST2/03905).

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