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arXiv:1206.0920 · AIP conference proceedings

A nonlinear dynamics approach to Bogoliubov excitations of Bose-Einstein condensates

Manuel Kreibich, Holger Cartarius, Jörg Main, G. Wunner

Published Jan 1, 2012• 1 citation
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Abstract

We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.

DOI
10.1063/1.4745583

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We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.
We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian stability matrix in a variational approach that maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r1/r1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibit a Rydberg-like structure, which can be explained by means of a quantum-defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussian functions combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates and the eigenfrequencies of the excitation modes are compared.
The nonlinear dynamics of self-bound Bose–Einstein matter waves under the action of an attractive nonlocal and a repulsive local interaction is analyzed by means of a time-dependent variational formalism. The mean-field model described by the Gross–Pitaevskii equation (GPE) is reduced to a single second-order conservative ordinary differential equation admitting a potential function. The stable fixed points and the linear oscillation frequencies of the breather modes are determined. The chosen nonlocal interactions are a Gaussian-shaped potential and a van der Waals-like potential. The variational solutions are compared with direct numerical simulations of the GPE, for each of these nonlocalities. The spatio-temporal frequency spectra of linear waves are also determined.
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We solve the time-dependent Gross-Pitaevskii equation by a variational ansatz to calculate the excitation spectrum of a Bose-Einstein condensate in a trap. The trial wave function is a Gaussian which allows an essentially analytical treatment of the problem. Our results reproduce numerical calculations over the whole range from small to large particle numbers, and agree exactly with the Stringari results in the strong interaction limit. Excellent agreement is obtained with the recent JILA experiment and predictions for the negative scattering length case are also made.
The decay of Bose-Einstein condensates from a metastable ground state into collapse due to macroscopic quantum tunneling is investigated using a semiclassical approximation to Feynman’s path integral formalism. Applying a variational ansatz of a single Gaussian to the wave function determined by the Gross-Pitaevskii equation, a special choice of the Gaussian width parameters yields a mean-field energy functional in Hamiltonian form. The temporal spatial extension of the condensate then is described in the picture of a particle moving in an external potential. In this picture the decay of the condensate wave function is investigated using the bounce trajectory method, accounting for the action of the tunneling orbit in imaginary time and quadratic fluctuations around it, the latter described by the Gelfand-Yaglom differential equation. Additionally, the tunneling formalism is used for describing the condensate wave function by a trial function of a superposition of several Gaussians. Using this ansatz a Hamiltonian form of the mean-field energy does not exist and the formalism of describing the tunneling process has to be extended. The bounce trajectory is described in the parameter space of the Gaussian width parameters and the equations of motion for the imaginary time evolution are obtained by applying a time-dependent variational principle. The action of the bounce trajectory is investigated as a function of the number of Gaussians taken into account and the contribution of the fluctuations is evaluated using the monodromy matrix.
We demonstrate parametric resonance in Bose-Einstein condensates (BECs) with attractive two-body interaction in a harmonic trap under parametric excitation by periodic modulation of the s-wave scattering length. We obtain nonlinear equations of motion for the widths of the condensate using a Gaussian variational ansatz for the Gross-Pitaevskii condensate wave function. We conduct both linear and nonlinear stability analyses for the equations of motion and find qualitative agreement, thus concluding that the stability of two equilibrium widths of a BEC might be inverted by parametric excitation.