Quantum Dynamics of Cold Atomic Gas with $SU(1,1)$ Symmetry

TL;DR

This paper explores the quantum dynamics of cold atomic gases with $SU(1,1)$ symmetry, expressing the evolution operator as an $SU(1,1)$ group element and visualizing dynamics on a Poincaré disk, with applications to Bose-Einstein condensation and Fermi gases.

Contribution

It introduces a method to represent the time evolution operator as an $SU(1,1)$ group element, enabling geometric visualization of quantum dynamics in cold atomic gases.

Findings

Trajectory visualization of Bose-Einstein condensation revival

Analysis of scale-invariant Fermi gas dynamics

Study of systems with time-dependent dispersion in oscillating lattices

Abstract

Motivated by recent advances in quantum dynamics, we investigate the dynamics of the system with $SU(1,1)$ symmetry. Instead of performing the time-ordered integral for the evolution operator of the time-dependent Hamiltonian, we show that the time evolution operator can be expressed as an $SU(1,1)$ group element. Since the $SU(1,1)$ group describes the "rotation" on a hyperbolic surface, the dynamics can be visualized on a Poincar\'e disk, a stereographic projection of the upper hyperboloid. As an example, we present the trajectory of the revival of Bose-Einstein condensation and that of the scale-invariant Fermi gas on the Poincar\'e disk. Further considering the quantum gas in the oscillating lattice, we also study the dynamics of the system with time-dependent single-particle dispersion. Our results are hopefully to be checked in current experiments.