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 $\ensuremath{\mu}$, 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 $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$. 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 $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$ because of the appearance of an adiabatic invariant. For $E/\ensuremath{\mu}$ 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 Poincar\'e surface of sections for harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps. For $E/\ensuremath{\mu}\ensuremath{\gg}1$ 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.
The Bogoliubov equations of the quasiparticle excitations in a weakly interacting trapped Bose condensate are solved in the WKB approximation in an isotropic harmonic trap, determining the discrete quasiparticle energies and wave functions by torus (Bohr-Sommerfeld) quantization of the integrable classical quasiparticle dynamics. The results are used to calculate the position and strengths of the peaks in the dynamic structure function, which can be observed by off-resonance inelastic light scattering.
Bose-Einstein condensation of ultra cold atoms is typically realized in magnetic traps which effectively lead to an axially symmetric harmonic potential. This letter shows that the spectrum of collective vibrational modes of a repulsive condensate in a prolate potential displays a defect known as quantum monodromy . The monodromy is analysed on the basis of the dynamics of quasiparticles. In terms of the quasiparticles the regime of collective modes or the so-called hydrodynamic regime is characterized through kinetic energies much smaller than the chemical potential. In this limit the classical dynamics of the quasiparticles is integrable. The monodromy is quantitatively described by a monodromy matrix that is calculated from classical actions.