Statistical permutation quantifiers in the classical transition of conservative-dissipative systems

The emergence of the classical world from the quantum domain is a topic of significant interest, both theoretically [1, 2, 3] and experimentally [4, 5, 6], as well as for its practical applications. The decoherence process [7], for instance, is closely linked to mesoscopic physics a field focusing on materials of intermediate scales. Mesoscopic systems, such as those used in nanotechnology, serve as practical examples [8, 9, 10, 11]. Additionally, semiclassical systems have been extensively employed over the years to address various physical problems [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

We have investigated systems that exhibit the coexistence of quantum and classical variables [24, 25, 26, 27, 28, 29, 30]. In these systems, the classical variables serve as a representation of the reservoir interacting with the quantum system.

Within this framework, we have explored the classical limit of quantum systems by employing a motion invariant associated with the Heisenberg Principle. Our approach includes dynamic analyses [24, 25, 26, 27, 28, 30] and the application of various tools from Information Theory (IT) [31, 32].

The classical limit of the system addressed in this work was previously examined in Ref. [30], utilizing Poincaré sections and analyzing the temporal evolution of key quantities. Both conservative and, for the first time, dissipative regimes were considered. In this study, we revisit the problem by employing Shannon entropy and two definitions of statistical complexity: the LMC complexity [33] and the Jensen-Shannon complexity [34]. To compute any information quantifier, a probability distribution function is required. For this purpose, we apply the Bandt-Pompe permutation based symbolic method [35], recognized for its efficiency in capturing key features of various time series [36, 37, 38, 39, 40, 41].

Our objective is to gain a deeper understanding and characterization of the process, uncovering details that cannot be obtained using dynamic tools alone.

We consider a Hamiltonian of the form

In this Hamiltonian, $\hat{x}$ and $\hat{p}$ are considered quantum operators while $A$ and $P_{A}$ canonically conjugated classic variables. $\hat{I}$ is the Identity operator. $\omega_{q}$ and $\omega_{cl}$ are frequencies and $e_{q}^{cl}$ a positive parameter positive [24]. The non-linear quantum-classical interaction appears through the term $e_{q}^{cl}A^{2}\hat{x}^{2}$ in (1). The semiclassical methodology that we use, is developed in the Appendix. The equations of motion that govern the system, as demonstrated in Ref. [30], are given by the following set of coupled non-linear equations

To study the classical limit of the system (LABEL:eq_semi), we need to compare with the solutions corresponding to the classical analogous of the Hamiltonian (1) [26]

By Hamilton’s equations we obtain (see Appendix)

We define $I$ as

$I$ is a motion-invariant for the systems (LABEL:eq_semi) in both regimens (conservative and dissipative) (see [24, 30]) and is related to the uncertainty principle describing the deviation with respect to classicity, a condition characterized by $I={x}^{2}\,{p}^{2}-L^{2}/4=0$ (trivial motion-invariant).
Eqs. (LABEL:eq_semi) do not explicitly depend upon $\hbar$ while mean values depend on it via the initial conditions through $I$ satisfying (5). The inequality (5) and its classical value, are crucial in our study of the classical limit $I\rightarrow 0$, as will be seen throughout the article. Focus now attention upon the relative energy $E_{r}$.

where $E=\left\langle H\right\rangle$ is the total energy. $E_{r}$ is also a motion invariant that verifies $E_{r}\geq 1$ due to the uncertainty principle. $E_{r}=1$ corresponds to the minimum value and represents the totally quantum case. $E_{r}$ is dimensionless. The LC is $I\rightarrow 0$, i.e.,

In the dissipative case we replace, in (6), $E$ by $E(0)$ [30]. Our main interest lies in the evolution of observable when $E_{r}$ grows from its minimum unity value $E_{r}=1$.

To calculate our Information quantifiers, it is necessary to have a probability distribution associated with the data series that one wants to analyze. In this case, of a time series of the semiclassical and classical systems. We obtain this probability distribution function (PDF) using the Bandt and Pompe method [35]. This methodology basically consists of a symbolization process that is described as follows: Let $\{x_{t}\}_{t=1,2,\cdots,N}$ be an arbitrary time series, the first step is to divide the time series into $n=N-\left(d-1\right)\tau$ overlapping partitions comprising $d>1$ observations separated by $\tau\geq 1$ time units. For given values of $d$ and $\tau$, each data partition can be represented by

where $p=1,2,\cdots,n$ is the partition index. The parameters $d$ and $\tau$ are the so-called embedding dimension and the embedding delay, respectively, the only two parameters of the Bandt and Pompe method. The original proposal presented by Bandt and Pompe was limited to $\tau=1$, this means that the data partitions comprised consecutive elements of time series, however, Cao et al. [42], Zunino et al. [43] and Pessa and Ribeiro [44], studied the embedding delay for values greater than 1.

Then, for each partition $w_{p}$ we evaluate the permutation $\pi_{p}=\left(r_{0},r_{1},\cdots,r_{d-1}\right)$ of the index numbers $\left(0,1,\cdots,d-1\right)$ which arranges the elements of $w_{p}$ in ascending order, that is, the permutation of the index numbers defined by the inequality $x_{p+r_{0}}\leq x_{p+r_{1}}\leq---\leq x_{p+r_{d-1}}$. In case of equality of values, the order of appearance of the elements of the partition is maintained, i.e., if $x_{p+r_{k-1}}=x_{p+rk}$, then $r_{k-1}<r_{k}$ for $k=1,\cdots,d-1$. After evaluating the permutation symbols associated with all data partitions, a symbolic sequence $\{\pi_{p}\}_{p=1,2,\cdots,n}$ is obtained.

The ordinal probability distribution $P=\{p_{i}\left(\Pi_{i}\right)\}_{i=1,2,\cdots,n_{\pi}}$ is the relative frequency of all possible permutations within the symbolic sequence

Here $\Pi_{i}$ represents each of the $n_{\pi}=d!$ different ordinal patterns.

The Bandt Pompe method has desirable features (i) simplicity, (ii) extremely fast calculation-process, and (iii) robustness. It is also invariant with respect to non-linear monotonous transformations. This method can be applied to any type of time series (regular, chaotic, noisy, or experimental) [31].

To solve the equations (LABEL:eq_semi) and (4) we choose the following parameter values: $\omega_{q}=\omega_{cl}=e_{q}^{cl}=1$. While for the initial conditions we take $\langle\hat{L}\rangle\left(0\right)=L(0)=A\left(0\right)=0$.

For the conservative system ($\eta=0$), we choose $E=0.6$. In the dissipative regimen we take the same value for the initial energy value $E(0)=0.6$. In both regimes we fixed this values, as we set $I\to 0$ ($E_{r}\to\infty$) in (6). In the dissipative cases is $\eta=0.05$.

We select $\langle\hat{x}^{2}\rangle\left(0\right)=E/\omega_{q}-0.98\sqrt{(E/\omega_{q})^{2}-I_{L}}$, where $I_{L}=I+\langle L\rangle(0)^{2}/4$. For the classic case is $x^{2}\left(0\right)=0.02E/\omega_{q}$, which results from taking the value $I=0$ in the previous expression. By appeal to (5) we see that $\langle\hat{p}^{2}\rangle(0)=\dfrac{I}{\langle\hat{x}^{2}\rangle}(0)$ ($p^{2}=0$ in the classical instance). In addition, from (1) we deduce $P_{A}(0)$ via

evaluating all quantities at the initial time. We have also an equivalent expression for the classical analogous system.

In the numerical calculation of the Shannon permutation entropy and the statistical complexities, the Python package ordpy developed by Arthur A. B. Pessa and Haroldo V. Ribeiro [44] was used. We have employed the values $d=5$ and $\tau=1$ for the dimensional embedding and embedding delay, respectively. The results have been conceptually confirmed using $d=6$.

The time series for the semiclassical case, have been constructed with the values of $\langle\hat{x}^{2}\rangle(t)$, each one for a different value of $E_{r}$ (or $I$). Also we have considered in the classical scenario ($I=0$), the unique series formed by the $x^{2}(t)$ values. In both cases to we have taken $N=20000$ points for each series. The condition $N>>d!=120$ is verified [31].

In all figures we observe low values of the information quantifiers, unlike what we have found for the chaotic Hamiltonian of Refs. [31, 32]. This feature are consistent with the regularity observed in the Poincaré Sections [30]. In detail, after the jump observed in the quasi-quantum zone between $E_{r}\approx 1$ and $E_{r}\approx 2.8$ (see below), one finds a very small band of variation of the values. However, the three regions of Refs. [31, 32] of the path to the classical analogous, continues being observed (quasi-quantum, transitional and classical).

It is convenient an analysis with additional tools such as information quantifiers, which, as mentioned, have proven efficiency for this type of problem.

In Fig.1 we plot the normalized Entropy $H$ vs. the relative energy $E_{r}$, for both conservative and dissipative regimes. In the conservative case (blue curve), between $E_{r}\approx 1$ and $E_{r}\approx 2.8$, we find out a zone, what we can call quasi-quantum. In the zone $E_{r}\approx 1$, with dynamic tools, quasi-periodic behavior has been observed (remember that $E_{r}=1$ is the lowest possible value of $E_{r}$ and corresponds to the fully quantum case). At $E_{r}\approx 2.8$ (first gray colored vertical dashed line), we have observed on a logarithmic scale, a drastic change (taking into account the mentioned small band of variation of $H$). The transition zone (mesoscopic) can be set between the values $E_{r}\approx 2.8$ and $E_{r}\approx 104.8$ and finally the convergence zone to the classical analogous value $H_{c}^{cl}=0.1768$, can be seen from $E_{r}\approx 104.8$ (second gray discontinuous vertical line).

The equivalent zones corresponding to the dissipative regime (orange curve) are: quasi-quantum from $E_{r}\approx 2.8$ to $E_{r}\approx 3.6$ (first plum colored vertical dashed line), the transitional from $E_{r}\approx 3.6$ to $E_{r}<123.0$, and the convergence zone to the analogous classic case from $E_{r}=123.0$ (second plum colored vertical dashed line). $H_{d}^{cl}=0.1720$, is the value of the entropy of the classical analogous in the dissipative case (remember that in this regime we replace in (6), $E$ by $E(0)$).

Additionally, the figure contains an inset in which the morphology of the curves in the transition zone is observed in detail. This region can be associated with a mesoscopic one, due to the important presence of quantum features. The zone of convergence to the totally classical system must correspond to a decoherence process [29].

In Figs. 2 we plot the two definitions of the statistical complexity vs. $E_{r}$. In both figures, one can clearly notice the three zones mentioned in Fig.1. These results are the same for both $C_{JS}$ and $C_{LMC}$. The blue curve corresponds to the conservative case and the orange curve to the dissipative one. The dashed vertical lines indicate the transition zone, gray for the conservative regime (between $2.8<E_{r}<104.8$) and plum for the dissipative case (from $E_{r}>3.6$ to $E_{r}<123.0$). Furthermore, in each graph there is a inset that basically consists of an increase in scale of the original graph, showing the transition zone. In Figure 2(a) we can see the statistical complexity of Jensen-Shannon $C_{JS}$. $C_{c}^{cl}=0.16401$ and $C_{d}^{cl}=0.16058$ are the statistical complexity values of the classical analogous of the conservative and dissipative regimes, respectively. Figure 2(b) presents the statistical complexity introduced by Lopez-Ruiz et al. $C_{LMC}$ vs $E_{r}$. The blue and the orange curves are associated to the conservative and dissipative system, respectively. In this case, $C_{c}^{cl}=0.13533$ and $C_{d}^{cl}=0.13426$ are the values of the statistical complexity of the classical analogous (conservative and dissipative regimen).

In this article we have analyzed the conservative and dissipative dynamics of the semiclassical Hamiltonian (1), which contains both quantum and classical variables. For this we have used information quantifiers, such as the Shannon entropy $H$ and the Statistical complexity. In the case of the latter, we analyze two versions, the pioneer one introduced by Lopez-Ruiz et al. $C_{LMC}$ [33, 45] and the Jensen Shannon complexity $C_{JS}$ [34]. The probability distribution to evaluate all these quantities was extracted from the time series using the Bandt and Pompe permutational method [35]. A methodology that has proven to be robust and reliable. In particular we have studied the classical limit of the system, that is, the convergence to the classical analogue of $H$ represented by Eq. (3) (only with classical variables), as a function of the relative energy $E_{r}$ (6), related to the Uncertainty Principle, with $E_{r}\to\infty$.

In [30] we had studied the same problem, but using dynamic tools such as Poincaré sections and projections of 3D Poincaré sections. The result found was the existence of three zones, quasi-quantum, transitional and finally one where the convergence to the results corresponding to the classical analogue was observed (classical zone). However, the boundaries between zones were diffuse, because the dynamics of the system is regular (see figures 5 and 6). The regularity of the dynamics is reflected in the figures of the present article, by their low values of all information quantifiers. Moreover, after the jump observed in the quasi-quantum zone, one finds a very small band of variation of the respective values.

It is seen in Figs. 1, 2, 3 and 4 that the Shannon permutation entropy $H$ and the statistical complexities $C_{LMC}$ and $C_{JS}$ confirm the mentioned three zones of the process (quasi-quantum, transitional or mesoscopic and that of decoherence, convergent to the classical analogous result). The corresponding intervals of $E_{r}$ are the same for the three quantifiers and can be synsetized by the transitional zone. In the conservative scenario $2.8<E_{r}<104.8$ and in the dissipative regime $3.6<E_{r}<123.0$.

We have found that the figures of $H$ and $C_{JS}$ have similar morphology but with a difference in scale. This is because the disequilibrium remains almost constant for all values of $E_{r}$ in both scenarios, conservative and dissipative. Disequilibrium corresponding to the $C_{LMC}$ complexity is also quasi-constant, but it has greater fluctuations around its mean value (conservative and dissipative regimes).

We conclude that the information quantifiers used in this work provide more details about the convergence process of the semiclassical system to the classical analogous, confirming the results obtained with the Poincaré sections used in Ref. [30], but with greater precision.