Quantum information entropies of ultra-cold atomic gases in a harmonic trap

TL;DR

This paper calculates the position and momentum space information entropies for ultra-cold atomic gases in harmonic traps, revealing universal entropy relations that depend on particle number and system type.

Contribution

It introduces a universal formula for the sum of entropies in trapped quantum gases, applicable to both Bose-Einstein condensates and Fermi gases, at zero temperature.

Findings

Entropy sum follows a universal form depending on atom number and trap dimension.

Results obey the entropic uncertainty relation by Beckner, Bialynicki-Birula, and Myceilski.

Distinct constants for bosonic and fermionic systems in the entropy formula.

Abstract

The position and momentum space information entropies of weakly interacting trapped atomic Bose-Einstein condensates and spin-polarized trapped atomic Fermi gases at absolute zero temperature are evaluated. We find that sum of the position and momentum space information entropies of these quantum systems contain $N$ atoms confined in a $D(\le 3)$-dimensional harmonic trap has a universal form as $ S_t^{(D)} = N(a D - b \ln N) $, where $a \simeq 2.332$ and $b = 2$ for interacting bosonic systems and $a \simeq 1.982$ and $b = 1$ for ideal fermionic systems. These results obey the entropic uncertainty relation given by the Beckner, Bialynicki-Birula and Myceilski.