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arXiv:1208.0991 · Journal of Physics B Atomic Molecular and Optical Physics

Geometric resonances in Bose–Einstein condensates with two- and three-body interactions

Hamid Al-Jibbouri, Ivana Vasić, Antun Balaž, Axel Pelster

Published Mar 7, 2013• 39 citations
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Abstract

We investigate geometric resonances in Bose-Einstein condensates by solving the underlying time-dependent Gross-Pitaevskii equation for systems with two- and three-body interactions in an axially-symmetric harmonic trap. To this end, we use a recently developed analytical method [Phys. Rev. A 84, 013618 (2011)], based on both a perturbative expansion and a Poincaré-Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters. By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances. Furthermore, we discuss in detail the stability of a Bose-Einstein condensate in the presence of an attractive two-body interaction and a repulsive three-body interaction. In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.

DOI
10.1088/0953-4075/46/6/065303

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We investigate geometric resonances in Bose-Einstein condensates by solving the underlying time-dependent Gross-Pitaevskii equation for systems with two- and three-body interactions in an axially-symmetric harmonic trap. To this end, we use a recently developed analytical method [Phys. Rev. A 84, 013618 (2011)], based on both a perturbative expansion and a Poincaré-Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters. By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances. Furthermore, we discuss in detail the stability of a Bose-Einstein condensate in the presence of an attractive two-body interaction and a repulsive three-body interaction. In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.
We investigate geometric resonances in Bose-Einstein condensates by solving the underlying time-dependent Gross-Pitaevskii (GP) equation for systems with two-and three-body interactions in an axially-symmetric harmonic trap.To this end, we use a recently developed analytical method [Vidanović I et al 2011 Phys.Rev. A 84 013618], based on both a perturbative expansion and a Poincaré-Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters.By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances.Furthermore, we discuss in detail the stability of a Bose-Einstein condensate in the presence of an attractive two-body interaction and a repulsive threebody interaction.In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.
We analytically and numerically study nonlinear dynamics in Bose–Einstein condensates (BECs) induced either by a harmonic modulation of the interaction or by the geometry of the trapping potential. To analytically describe BEC dynamics, we use a perturbative expansion based on the Poincaré–Lindstedt analysis of a Gaussian variational ansatz, whereas in the numerical approach we use numerical solutions of both a variational system of equations and the full time-dependent Gross–Pitaevskii equation. The harmonic modulation of the atomic s-wave scattering length of a BEC of 7 Li was achieved recently via Feshbach resonance, and such a modulation leads to a number of nonlinear effects, which we describe within our approach: mode coupling, higher harmonics generation and significant shifts in the frequencies of collective modes. In addition to the strength of atomic interactions, the geometry of the trapping potential is another key factor for the dynamics of the condensate, as well as for its collective modes. The asymmetry of the confining potential leads to important nonlinear effects, including resonances in the frequencies of collective modes of the condensate. We study in detail such geometric resonances and derive explicit analytic results for frequency shifts for the case of an axially symmetric condensate with two- and three-body interactions. Analytically obtained results are verified by extensive numerical simulations.
The dynamics of weakly interacting three-dimensional Bose-Einstein condensates (BECs), trapped in external axially symmetric plus anharmonic distortion potential are studied. Within a variational approach and time-dependent Gross-Pitaevskii equation, the coupled condensate width equations are derived. By modulating anharmonic distortion of the trapping potential, nonlinear features are studied numerically and illustrated analytically, such as mode coupling of oscillation modes, and resonances. Furthermore, the stability of attractive interaction BEC in both repulsive and attractive anharmonic distortion is examined. We demonstrate that a small repulsive and attractive anharmonic distortion is effective in reducing (extending) the condensate stability region since it decreases (increases) the critical number of atoms in the trapping potential.
The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.
This paper investigates the collective excitation and stability of low-dimensional Bose–Einstein condensates with two- and three-body interactions by the variational analysis of the time-dependent Gross–Pitaevskii–Ginzburg equation. The spectrum of the low-energy excitation and the effective potential for the width of the condensate are obtained. The results show that: (i) the repulsive two-body interaction among atoms makes the frequency red-shifted for the internal excitation and the repulsive or attractive three-body interaction always makes it blue-shifted; (ii) the region for the existence of the stable bound states is obtained by identifying the critical value of the two- and three-body interactions.
We study the stabilization of a trapless Bose-Einstein condensate by analysing the mean-field Gross-Pitaevskii equation with attractive two- and three-body interactions through both analytical and numerical methods. Using the variational method we show that there is an enhancement of the condensate stability due to the inclusion of a three-body interaction in addition to the two-body interaction. We also study the stability of the condensates in the presence of the time-varying three-body interaction. Finally we confirm the stabilization of a trapless condensate from numerical simulation.