Dynamical features of Shannon information entropy of bosonic cloud in a tight trap

TL;DR

This paper investigates the behavior and dynamics of Shannon information entropy in trapped bosonic clouds, revealing how entropy relates to stability, nonlinearity, and trap tightness, with implications for understanding quantum correlations.

Contribution

It provides a detailed analysis of Shannon entropy in bosonic condensates, including its functional form, stability criteria, and dynamic behavior under varying interactions and trap conditions.

Findings

Total entropy follows a logarithmic relation with particle number.

Entropy diverges at the collapse point for attractive interactions.

Entropy dynamics remain consistent with uncertainty principles despite nonlinearity.

Abstract

We calculate Shannon information entropy of trapped interacting bosons in both the position and momentum spaces, $S_r$ and $S_k$ respectively. The total entropy maintains the fuctional form $S=a + b \ln N$ for repulsive bosons. At the noninteracting limit the lower bound of entropic uncertainty relation is also satisfied whereas the diverging behavior of $S_r$ and $S_k$ at the critical point of collapse for attractive condensate accurately calculates the stability factor. Next we study the dynamics of Shannon information entropy with varying interparticle potential. We numerically solve the time-dependent Gross-Pitaevskii equation and study the influence of increasing nonlinearity in the dynamics of entropy uncertainty relation (EUR). We observe that for small nonlinearity the dynamics is regular. With increase in nonlinearity although Shannon entropy shows large variation in amplitude of the oscillation, the EUR is maintained throughout time for all cases and it confirms its generality. We also study the dynamics in a very tight trap when the condensate becomes highly correlated and strongly inhomogeneous. Time evolution of total entropy exhibits aperiodic and fluctuating nature in very tight trap. We also calculate Landsberg's order parameter for various interaction strengths which supports earlier observation that entropy and order are decoupled.