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Journal of Physics A Mathematical and Theoretical

Nonlinear from linear states in two-component Bose–Einstein condensates

Georgia Karali, P. G. Kevrekidis, Nikolaos K. Efremidis

Published Dec 19, 2008• 2 citations
Publisher

Abstract

In this work, we consider the extension of linear eigenmodes of the harmonic oscillator potential into nonlinear states, for the case of two-component Gross–Pitaevskii equations with a parabolic potential, motivated by the context of two interacting hyperfine states of 87 Rb in Bose–Einstein condensates. In particular, we establish that nonlinear continuations of various eigenmode combinations are possible and corroborate this analytical finding with numerical computations for the lowest few eigenmode combinations involving the ground state and the first two excited states. A multitude of nonlinear states can be constructed in this way, some of which spontaneously deform, as the interactions become stronger, into previously obtained nonlinear eigenstates. The Bogolyubov–de Gennes analysis of the excitations on top of such states illustrates that some of them may become unstable beyond a critical threshold (of the chemical potentials associated with the states), while others may be stable within the entire range of chemical potentials considered herein. When the modes are found to be unstable, their evolution is followed, leading to interesting dynamical effects such as spontaneous symmetry breaking or oscillatory growth.

DOI
10.1088/1751-8113/42/4/045206

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In this work, we consider the extension of linear eigenmodes of the harmonic oscillator potential into nonlinear states, for the case of two-component Gross–Pitaevskii equations with a parabolic potential, motivated by the context of two interacting hyperfine states of 87 Rb in Bose–Einstein condensates. In particular, we establish that nonlinear continuations of various eigenmode combinations are possible and corroborate this analytical finding with numerical computations for the lowest few eigenmode combinations involving the ground state and the first two excited states. A multitude of nonlinear states can be constructed in this way, some of which spontaneously deform, as the interactions become stronger, into previously obtained nonlinear eigenstates. The Bogolyubov–de Gennes analysis of the excitations on top of such states illustrates that some of them may become unstable beyond a critical threshold (of the chemical potentials associated with the states), while others may be stable within the entire range of chemical potentials considered herein. When the modes are found to be unstable, their evolution is followed, leading to interesting dynamical effects such as spontaneous symmetry breaking or oscillatory growth.
Abstract We systematically construct vector solitary waves in harmonically trapped one-dimensional two-component Bose–Einstein condensates with unequal dispersion coefficients by a numerical continuation in chemical potentials from the respective analytic low-density linear limits to the high-density nonlinear Thomas-Fermi regime. The main feature of the linear states herein is that the component with the larger quantum number has instead a smaller linear eigenenergy, enabled by suitable unequal dispersion coefficients, leading to new series of solutions compared with the states similarly obtained in the equal dispersion setting. Particularly, the lowest-lying series gives the well-known dark-anti-dark waves, and the second series yields the dark-multi-dark states, and the following series become progressively more complex in their wave structures. The Bogoliubov-de Gennes spectra analysis shows that most of these states bear unstable modes, but they can be long-lived and remarkably all of them can be fully stabilized in suitable parameter regimes.
We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.
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{n}_{r}=0,1nr​=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 mmm, 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=0m=0m=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{n}_{r}=0nr​=0 and m=0m=0m=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.
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.
Abstract In this work, we explore systematically various SO(2)-rotation-induced multiple dark–dark (DD) soliton breathing patterns obtained from stationary and spectrally stable multiple dark–bright (DB) and DD waveforms in trapped one-dimensional, two-component atomic Bose–Einstein condensates. The stationary states stemming from the associated linear limits (as the eigenfunctions of the quantum harmonic oscillator problem) are parametrically continued to the nonlinear regimes by varying the respective chemical potentials, i.e. from the low-density linear limits to the high-density Thomas–Fermi (TF) regimes. We perform a Bogolyubov–de Gennes spectral stability analysis to identify stable parametric regimes of these states, finding a wide range of stability intervals in the TF regimes for all of the states considered herein. Upon applying an SO(2)-rotation to stable steady states, one-, two-, three-, four-, and many DD soliton breathing patterns are observed in the numerical simulations. Furthermore, analytic solutions up to three DB solitons in the homogeneous setting, and three-component systems are also investigated.