Pairwise and higher-order statistical correlations in excited states of quantum oscillator systems

Abstract

Pairwise and higher-order statistical correlations are examined in excited states of quantum oscillator systems, in position, and in momentum space. Measures from information theory, and those based on higher-order moments of the distribution, are used to quantify the correlations. Transition points are observed in the pairwise measures as the intensity of the interaction potential is varied. The presence of these points in a particular space, is governed by the symmetry of the wave function, and the attractive or repulsive nature of the interaction potentials. Crossover points in the higher-order measures are also present, where the interaction information transits from positive to negative values as the intensity of the interaction potential increases. The interpretation is that the dominance of synergic interactions depends on the intensity of the potential. The appearance of these crossover points in a particular representation is determined by the nature of the interaction potential, and not by the symmetry of the wave function. The magnitudes of the interaction potentials at these transition and crossover points, in each space, are shown to be a consequence of a relation which establishes a functional equivalence between the wave functions in each representation. Differences in the correlation characteristics of excited and ground states are discussed.

Appendix

The similarity in behaviour between mutual information and cokurtoses measures at the two and three variable levels has been reported for the $\left| 000\right\rangle$ symmetric and $\left| 001\right\rangle$ antisymmetric ground states in position space [51]. Mutual information is taken from information theory while the cokurtosis comes from a moments-based approach. Although distinct, both are thought to measure the nonlinear correlations. The question remains, if the similarity that was observed in position space is carried through to momentum space. Thus, we illustrate the behaviours of these measures in momentum space for the ground states. This information supplements that presented for the excited states in the body of this work.

Figure 12 presents a comparison of the pairwise measures. First, the similarity between the $I_p$ and $Cok(p_1,p_2)$ measures is striking in all cases. The minimum that is present in $I_p$ in the $\left| 001\right\rangle$ antisymmetric state with repulsive potential, is also present in the $Cok(p_1,p_2)$ measure. In this state, $\tau _p$ is negative-valued at smaller $\lambda _-$, and transits to positive values as $\lambda _-$ increases. This transition point is also detected by the minima in $I_p$ and in $Cok(p_1,p_2)$, which occur in the same region. The position of these minima are shifted to smaller values as compared to the transition point.

Figure 13 presents the comparison of triple-wise and higher-order measures. All $I_3^p$ measures are consistent in behaviour with $Cok(p_1,p_2,p_3)$, except for the antisymmetric $\left| 001\right\rangle$ state with attractive potential, whose cokurtosis displays a maximum. Note that the presence of the minimum in $I_3^p$ for the $\left| 001\right\rangle$ state with repulsive potential is also captured by a corresponding minimum in $Cok(p_1,p_2,p_3)$ in the same region. There is also a striking resemblance between the higher-order interaction information, $I_p^3$, and the $\eta _p$ measure based on the cokurtoses. Note that $\eta _p$ captures the positive to negative transition in $I_p^3$, at larger $\lambda _-$, in the $\left| 001\right\rangle$ state with repulsive potential. We emphasize that the values of $\lambda _-$ at this transition is distinct for each measure.

Fig. 12

Behaviour of pairwise correlation measures vs. the intensity of an attractive or repulsive interaction potential in momentum space, for the $\left| 001\right\rangle$ state (first, third rows), and the $\left| 000\right\rangle$ one (second, fourth rows). The dashed magenta line corresponds to the limiting value of the repulsive potential, while the black dashed line ($\lambda _-=0.5$) corresponds to the transition point in the correlation coefficient

Fig. 13

Behaviour of triple-wise correlation measures vs. the intensity of an attractive or repulsive interaction potential in momentum space, for the $\left| 001\right\rangle$ state (first, third rows) and the $\left| 000\right\rangle$ one (second, fourth rows). The dashed magenta line corresponds to the limiting value of the repulsive potential