Classical quasi-particle dynamics in trapped Bose condensates

TL;DR

This paper investigates the classical dynamics of quasi-particles in trapped Bose-Einstein condensates, revealing integrable and chaotic regimes depending on energy and trap anisotropy, with implications for understanding collective excitations.

Contribution

It provides a detailed analysis of quasi-particle dynamics in various trap geometries, identifying conditions for integrability and chaos, and introduces nearly conserved phase-space functions at low energies.

Findings

Classical motion is integrable in isotropic traps.

Anisotropic traps exhibit nonintegrable, chaotic dynamics.

Low-energy quasi-particle dynamics are asymptotically integrable due to additional invariants.

Abstract

The dynamics of quasi-particles in repulsive Bose condensates in a harmonic trap is studied in the classical limit. In isotropic traps the classical motion is integrable and separable in spherical coordinates. In anisotropic traps the classical dynamics is found, in general, to be nonintegrable. For quasi-particle energies E much smaller than thechemical potential, besides the conserved quasi-particle energy, we identify two additional nearly conserved phase-space functions. These render the dynamics inside the condensate (collective dynamics) integrable asymptotically for E/chemical potential very small. However, there coexists at the same energy a dynamics confined to the surface of the condensate, which is governed by a classical Hartree-Fock Hamiltonian. We find that also this dynamics becomes integrable for E/chemical potential very small, because of the appearance of an adiabatic invariant. For E/chemical potential of order 1 a large portion of the phase-space supports chaotic motion, both, for the Bogoliubov Hamiltonian and its Hartree-Fock approximant. To exemplify this we exhibit Poincar\'e surface of sections for harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps. For E/chemical potential very large the dynamics is again governed by the Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes quasi-integrable because the remaining small chaotic components in phase space are tightly confined by tori.