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arXiv:0809.2007 · AIP conference proceedings

Nonlinear Dynamics of Bose-Einstein Condensates with Long-Range Interactions

G. Wunner, H. Cartarius, T. Fabčič, P. Köberle, Jörg Main, T. Schwidder, Marko Robnik, Valery Romanovski

Published Jan 1, 2008• 2 citations
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Abstract

The motto of this paper is: Let’s face Bose‐Einstein condensation through nonlinear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose‐Einstein condensates via the time‐dependent Gross‐Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonlinear dynamics. We consider in particular cold quantum gases where long‐range interactions between the neutral atoms are present, in addition to the conventional short‐range contact interaction, viz. gravity‐like interactions, and dipole‐dipole interactions. The results obtained serve as a useful guide in the search for nonlinear dynamics effects in numerically exact quantum calculations for Bose‐Einstein condensates. A main result is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which could be checked experimentally.

DOI
10.1063/1.3046264

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The motto of this paper is: Let’s face Bose‐Einstein condensation through nonlinear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose‐Einstein condensates via the time‐dependent Gross‐Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonlinear dynamics. We consider in particular cold quantum gases where long‐range interactions between the neutral atoms are present, in addition to the conventional short‐range contact interaction, viz. gravity‐like interactions, and dipole‐dipole interactions. The results obtained serve as a useful guide in the search for nonlinear dynamics effects in numerically exact quantum calculations for Bose‐Einstein condensates. A main result is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which could be checked experimentally.
Abstract. The motto of this paper is: Let's face Bose-Einstein condensation through nonUnear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose-Einstein condensates via the time-dependent Gross-Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonUnear dynamics. We consider in particular cold quantum gases where long-range interactions between the neutral atoms are present, in addition to the conventional short-range contact interaction, viz. gravity-Uke interactions, and dipole-dipole interactions. The residts obtained serve as a useful giude in the search for nonlinear dynamics effects in numerically exact quantum calcinations for Bose-Einstein condensates. A main residt is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which coidd be checked experimentally. Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, nonlinear dynamics effects PACS: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky 1.
In this paper, we study the response of a Bose–Einstein condensate with strong dipole–dipole atomic interactions to periodically varying perturbation. The dynamics is governed by the Gross–Pitaevskii equation with additional nonlinear term, corresponding to a nonlocal dipolar interactions. The mathematical model, based on the variational approximation, has been developed and applied to parametric excitation of the condensate due to periodically varying coefficient of nonlocal nonlinearity. The model predicts the waveform of solitons in dipolar condensates and describes their small amplitude dynamics quite accurately. Theoretical predictions are verified by numerical simulations of the nonlocal Gross–Pitaevskii equation and good agreement between them is found. The results can lead to better understanding of the properties of ultra-cold quantum gases, such as 52 Cr , 164 Dy and 168 Er , where the long-range dipolar atomic interactions dominate the usual contact interactions.
We apply a variational technique to solve the time-dependent Gross–Pitaevskii equation for Bose–Einstein condensates in which an additional dipole–dipole interaction between the atoms is present with the goal of modelling the dynamics of such condensates. We show that universal stability thresholds for the collapse of the condensates correspond to bifurcation points where always two stationary solutions of the Gross–Pitaevskii equation disappear in a tangent bifurcation, one dynamically stable and the other unstable. We point out that the thresholds also correspond to 'exceptional points', i.e. branching singularities of the Hamiltonian. We analyse the dynamics of excited condensate wave functions via Poincaré surfaces of section for the condensate parameters and find both regular and chaotic motion, corresponding to (quasi-) periodically oscillating and irregularly fluctuating condensates, respectively. Stable islands are found to persist up to energies well above the saddle point of the mean-field energy, alongside collapsing modes. The results are applicable when the shape of the condensate is axisymmetric.
The variational method of coupled Gaussian functions is applied to Bose-Einstein condensates with long-range interactions. The time dependence of the condensate is described by dynamical equations for the variational parameters. We present the method and analytically derive the dynamical equations from the time-dependent Gross-Pitaevskii equation. The stability of the solutions is investigated using methods of nonlinear dynamics. The concept presented in this article will be applied to Bose-Einstein condensates with monopolar 1/r1/r1/r and dipolar 1/r31/{r}^{3}1/r3 interaction in the subsequent article [S. Rau et al., Phys. Rev. A 82, 023611 (2010)], where we will present a wealth of phenomena obtained using the ansatz with coupled Gaussian functions.
The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r1/r1/r interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses.
Bose-Einstein condensates with an attractive 1/r1/r1/r interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A 82, 023610 (2010)]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both stable and unstable states of the Gross-Pitaevskii equation and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov--de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.