International Journal of Bifurcation and Chaos

SPATIAL RESONANCE OVERLAP IN BOSE–EINSTEIN CONDENSATES IN SUPERLATTICE POTENTIALS

Vivien P. Chua, Mason Alexander Porter

Published Apr 1, 2006• 6 citations

Abstract

We employ Chirikov's overlap criterion to investigate interactions between subharmonic resonances of coherent structure solutions of the Gross–Pitaveskii (GP) equation governing the mean-field dynamics of cigar-shaped Bose–Einstein condensates in optical superlattices. We apply a standing wave ansatz to the GP equation to obtain a parametrically forced Duffing equation describing the BEC's spatial dynamics. We then investigate analytically the dependence of spatial resonances on the depth of the superlattice potential, deriving an order-of-magnitude estimate for the critical depth at which spatial resonances with respect to different lattice harmonics first overlap. We also derive a formula for the size of resonance zones and examine changes in our estimates as the relative superlattice amplitudes corresponding to the different harmonics are adjusted. We investigate the onset of global chaos and support our analytical work with numerical simulations.

DOI: 10.1142/s0218127406015222

Related papers

Bose-Einstein Condensates in Superlattices

Mason Alexander Porter, P. G. Kevrekidis

Abstract. We consider the Gross–Pitaevskii (GP) equation in the presence of periodic and quasi-periodic su-perlattices to study cigar-shaped Bose–Einstein condensates (BECs) in such potentials. We examine spatially extended wavefunctions in the form of modulated amplitude waves (MAWs). With a coher-ent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply second-order multiple scale perturbation theory to study harmonic resonances with respect to a single lattice substructure as well as ultrasubharmonic resonances that result from interactions of both substructures of the superlattice. In each case, we determine the resulting system’s equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding wavefunctions by direct simulations of the GP equation, identify-ing them as typically stable solutions of the model. We then study subharmonic resonances using Hamiltonian perturbation theory, tracing robust spatio-temporally periodic patterns. Key words. Bose–Einstein condensates, multiple scale perturbation theory, Hamiltonian systems

Jan 1, 200518 citations

Parametric and geometric resonances of collective oscillation modes in Bose–Einstein condensates

Ivana Vidanović, Hamid Al-Jibbouri, Antun Balaž +1 more

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.

Physica ScriptaApr 27, 201222 citations

Feshbach resonance management of Bose-Einstein condensates in optical lattices

Mason Alexander Porter, Marina Chugunova, Dmitry E. Pelinovsky

We analyze gap solitons in trapped Bose-Einstein condensates (BECs) in optical lattice potentials under Feshbach resonance management. Starting with an averaged Gross-Pitaevsky equation with a periodic potential, we employ an envelope-wave approximation to derive coupled-mode equations describing the slow BEC dynamics in the first spectral gap of the optical lattice. We construct exact analytical formulas describing gap soliton solutions and examine their spectral stability using the Chebyshev interpolation method. We show that these gap solitons are unstable far from the threshold of local bifurcation and that the instability results in the distortion of their shape. We also predict the threshold of the power of gap solitons near the local bifurcation limit.

arXiv:nlin/0507025Physical Review ESep 15, 200621 citations

Controlling Chaos Probability of a Bose–Einstein Condensate in a Weak Optical Superlattice

Xu Jun, Xiaobing Luo

The spatial chaos probability of a Bose–Einstein condensate perturbed by a weak optical superlattice is studied. It is demonstrated that the spatial chaotic solution appears with a certain probability in a given parameter region under a random boundary condition. The effects of the lattice depths and wave vectors on the chaos probability are illustrated, and different regions associated with different chaos probabilities are found. This suggests a feasible scheme for suppressing and strengthening chaos by adjusting the optical superlattice experimentally.

Chinese Physics LettersMar 31, 20094 citations

Spatial structure of a collisionally inhomogeneous Bose-Einstein condensate

Fei Li, Dongxia Zhang, Shiguang Rong +1 more

The spatial structure of a collisionally inhomogeneous Bose-Einstein condensate (BEC) in an optical lattice is studied. A spatially dependent current with an explicit analytic expression is found in the case with a spatially dependent BEC phase. The oscillating amplitude of the current can be adjusted by a Feshbach resonance, and the intensity of the current depends heavily on the initial and boundary conditions. Increasing the oscillating amplitude of the current can force the system to pass from a single-periodic spatial structure into a very complex state. But in the case with a constant phase, the spatially dependent current disappears and the Melnikov chaotic criterion is obtained via a perturbative analysis in the presence of a weak optical lattice potential. Numerical simulations show that a strong optical lattice potential can lead BEC atoms to a state with a chaotic spatial distribution via a quasiperiodic route.

Journal of Experimental and Theoretical PhysicsNov 1, 20136 citations

Collective Excitations in Bose-Einstein Condensates

Hamid Al-Jibbouri

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.

Jan 1, 20130 citations