Classical quasiparticle dynamics in trapped Bose condensates

The dynamics of quasiparticles 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 quasiparticle energies $E$ much smaller than the chemical potential $\mu$, besides the conserved quasiparticle energy, we identify two additional nearly conserved phase-space functions. These render the dynamics inside the condensate (collective dynamics) integrable asymptotically for $E/\mu \to 0$. 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 $E/\mu \to 0$ because of the appearance of an adiabatic invariant. For $E/\mu$ 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 Poincaré surface of sections for harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps. For $E/\mu \gg 1$ the dynamics is again governed by the Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes quasiintegrable because the remaining small chaotic components in phase space are tightly confined by tori.