Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with
nr=0,1, and 2 radial nodes, respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge
m, i.e., vorticity. It is found that for repulsive interatomic interactions only the ground state is linearly stable throughout the parameter range examined. Furthermore, this is true for topological charges
m=0 or 1; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing (attractive) case the ground state with
nr=0 and
m=0 can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.
We study the linearized Bogoliubov excitation spectrum of infinitely long anharmonically trapped Bose–Einstein condensates, with the aim of overcoming inhomogeneous broadening. We compare the Bogoliubov spectrum of a harmonic trap with that of a theoretical flat-bottom trap and find a significant reduction in the inhomogeneous broadening of the lineshape of Bogoliubov excitations. While the Bragg excitation spectrum for a condensate in a harmonic trap supports a number of radial modes, the flat trap is found to significantly support just one mode. We also study the excitation spectrum of realistic anharmonic traps with potentials of finite power dependence on the radial coordinate. We observe a correlation between the number of radial modes and the number of bound states in the effective potential of the quasi-particles. Finally, we compare a full numerical Gross–Pitaevskii simulation of a finite-length condensate to our model of infinite, linearized Gross–Pitaevskii excitations. We conclude that our model captures the essential physics.
We study the linearized Bogoliubov excitation spectrum of infinitely long anharmonically trapped Bose-Einstein condensates, with the aim of overcoming inhomogeneous broadening. We compare the Bogoliubov spectrum of a harmonic trap with that of a theoretical flat-bottom trap and find a dramatic reduction in the inhomogeneous broadening of the lineshape of Bogoliubov excitations. While the Bragg excitation spectrum for a condensate in a harmonic trap supports a number of radial modes, the flat trap is found to significantly support just one mode. We also study the excitation spectrum of realistic anharmonic traps with potentials of finite power dependence on the radial coordinate. We observe a correlation between the number of radial modes and the number of bound states in the effective potential of the quasi-particles. Finally we compare a full numerical Gross-Pitaevskii simulation of a finite-length condensate to our model of infinite, linearized Gross-Pitaevskii excitations. We conclude that our model captures the essential physics.
We study the stability of Bose-Einstein condensate (BEC) with repulsive and attractive interatomic interactions in a harmonic plus quartic trap $[V(r)=\frac{1}{2}m{\ensuremath{\omega}}^{2}{r}^{2}+\ensuremath{\lambda}{r}^{4}]$. We find that BEC with repulsive interaction becomes more stable for $\ensuremath{\lambda}>0$. For $\ensuremath{\lambda}<0$, we observe a different metastability of the condensate even when the scattering length is positive. For BEC with attractive interaction, we observe dramatic change in the stability factor
ahoAcr∣asc∣ due to the anharmonicity. The frequencies of the collective excitations are also modified by the atomic interaction and by the anharmonicity.
We study the instability of collective excitations of a three-dimensional Bose-Einstein condensate with repulsive and attractive interactions in a shallow trap designed as a quadratic plus a quartic potential. By using a correlated many-body theory, we determine the excitation modes and probe the critical behavior of collective modes, having a crucial dependence on the anharmonic parameter. We examine the power-law behavior of monopole frequency near criticality. In Gross-Pitaevskii variational treatment [Phys. Rev. Lett. 80, 1576 (1998)] the power-law exponent is determined as one-fourth power of $(1\ensuremath{-}\frac{A}{{A}_{\mathit{cr}}})$,
A is the number of condensate atoms and
Acr is the critical number near collapse. We observe that the power-law exponent becomes
61 in our calculation for the pure harmonic trap and it becomes
71, for traps with a small anharmonic distortion. However for large anharmonicity the power law breaks down.