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Physical Review A

Dynamics of matter-wave solitons in harmonic traps with flashing optical lattices

Tristram J. Alexander, Kimberley Anne Heenan, Mario Salerno, Elena A. Ostrovskaya

Published Jun 29, 2012• 11 citations
Publisher

Abstract

We study, analytically and numerically, the dynamics of a matter-wave soliton formed by a Bose-Einstein condensate with an attractive interatomic interaction confined in a one-dimensional harmonic trap and subjected to a flashing weak periodic potential. Within the framework of the Gross-Pitaevski equation, we show that efficient energy transfer to the soliton occurs within two distinct frequency regimes: (a) on resonance with the trapping frequency and (b) on resonance with the internal mode of the soliton. We find that the former regime is well described within the soliton-as-particle approximation, while the latter relies on the extended nature of the localized state and involves strong excitations of the width coupled to translational motion.

DOI
10.1103/physreva.85.063626

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We study, analytically and numerically, the dynamics of a matter-wave soliton formed by a Bose-Einstein condensate with an attractive interatomic interaction confined in a one-dimensional harmonic trap and subjected to a flashing weak periodic potential. Within the framework of the Gross-Pitaevski equation, we show that efficient energy transfer to the soliton occurs within two distinct frequency regimes: (a) on resonance with the trapping frequency and (b) on resonance with the internal mode of the soliton. We find that the former regime is well described within the soliton-as-particle approximation, while the latter relies on the extended nature of the localized state and involves strong excitations of the width coupled to translational motion.
In this thesis we study, analytically and numerically, the dynamics of a bright matter-wave soliton formed from a Bose-Einstein condensate with attractive interatomic interactions, confined by a one dimensional harmonic or double-well trapping potential, and subject to weak periodic driving. Such systems have potential applications in metrology, specifically in the use of atom interferometers for high precision inertial and gravitational field sensing. The localised and dynamically robust nature of matter-wave solitons may help overcome limitations such as wavepacket spreading in current atom interferometry experiments. A typical matter-wave interferometer involves a condensate trapped in a double-well potential. Being able to manipulate the tunnelling behaviour of a matter-wave soliton in potentials such as a double-well would thus give greater control in setting up these experiments. This also has a bearing upon the broader problem of transport of matter-wave solitons across potential barriers. These are questions which are largely unanswered in the current literature. We understand the tunnelling behaviour of a matter-wave soliton in a driven double well potential by first studying the fundamental mechanisms by which energy is transferred to a soliton in a driven harmonic potential. In this way our work also contributes more generally to the problem of transport in driven potentials. Within the mean field approximation, we show that efficient energy transfer to the soliton occurs in two frequency regimes: on resonance with the trapping frequency and on resonance with the internal mode frequency of the soliton. The former regime is well described by modelling the soliton effectively as a rigid particle, and in this regime energy transfer occurs due to parametric driving. In the latter regime, the extended nature of the soliton is important and energy transfer occurs due to internal mode coupling: a coupling between the width oscillations of the soliton and its translational motion. We find that in this internal mode coupling regime the soliton is unstable to decay. We thus show that effective particle regime is the more promising for efficiently exciting a bright matter-wave soliton to tunnel across a potential barrier, while preserving its form.
We study the dynamics of bright solitons formed in a Bose-Einstein condensate with attractive atomic interactions perturbed by a weak bichromatic optical lattice potential. The lattice depth is a biperiodic function of time with a zero mean, which realizes a flashing ratchet for matter-wave solitons. We find that the average velocity of a soliton and the soliton current induced by the ratchet depend on the number of atoms in the soliton. As a consequence, soliton transport can be induced through scattering of different solitons. In the regime when matter-wave solitons are narrow compared to the lattice period the dynamics is well described by the effective Hamiltonian theory.
The mean-field dynamics of Bose-Einstein condensates loaded in an optical lattice, confined by a parabolic potential, and subjected to a change of scattering length by means of the Feshbach resonance is considered. The system is described by the Gross-Pitaevskii (GP) equation with varying nonlinearity, which in a number of cases is reduced to one-dimensional (1D) perturbed nonlinear Schr\"odinger (NLS) equations, the particular form of which depends on the relation among the parameters of the problem. We analytically describe the adiabatic dynamics of periodic solutions of the respective NLS equations, provide a numerical study of 1D models confined by a parabolic trap, and carry out numerical simulations of the matter-wave dynamics within the framework of the radially symmetric 3D GP equation. Special attention is paid to processes of generation of trains of bright and dark matter solitons from initially periodic waves. The results of the 1D approximation are compared with direct numerical simulation of the original 3D GP eqation, showing remarkable coincidence for definite regions of parameters.
The dynamics of a bright matter wave soliton in a quasi one-dimensional Bose–Einstein condensate (BEC) with a periodically rapidly varying time trap is considered. The governing equation is based on averaging the fast modulations of the Gross–Pitaevskii (GP) equation. This equation has the form of a GP equation with an effective potential of a more complicated structure than an unperturbed trap. In the case of an inverted (expulsive) quadratic trap corresponding to an unstable GP equation, the effective potential can be stable. For the bounded space trap potential it is showed that bifurcation exists, i.e. the single-well potential bifurcates to the triple-well effective potential. The stabilization of a BEC cloud on-site state in the temporary modulated optical lattice is found. This phenomenon is analogous to the Kapitza stabilization of an inverted pendulum. The analytical predictions of the averaged GP equation are confirmed by numerical simulations of the full GP equation with rapid perturbations.
Dynamics and stability of solitons in two-dimensional (2D) Bose-Einstein condensates (BEC), with one-dimensional (1D) conservative plus dissipative nonlinear optical lattices, are investigated. In the case of focusing media (with attractive atomic systems), the collapse of the wave packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in focusing 2D media with 1D periodic nonlinearity. In the defocusing media (repulsive BEC case) with harmonic trap in one direction and nonlinear optical lattice in the other direction, the stable soliton can exist. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.
We consider the ground state of an attractively interacting atomic Bose-Einstein condensate in a prolate, cylindrically symmetric harmonic trap. If a true quasi-one-dimensional limit is realized, then for sufficiently weak axial trapping this ground state takes the form of a bright soliton solution of the nonlinear Schr\"odinger equation. Using analytic variational and highly accurate numerical solutions of the Gross-Pitaevskii equation, we systematically and quantitatively assess how solitonlike this ground state is, over a wide range of trap and interaction strengths. Our analysis reveals that the regime in which the ground state is highly solitonlike is significantly restricted and occurs only for experimentally challenging trap anisotropies. This result and our broader identification of regimes in which the ground state is well approximated by our simple analytic variational solution are relevant to a range of potential experiments involving attractively interacting Bose-Einstein condensates.