Attractive Bose-Einstein Condensate Research Articles

Integrable NLS equation with time-dependent nonlinear coefficient and self-similar attractive BEC

We investigate the nonlinear Schrödinger equation with a time-dependent nonlinear coefficient. By means of Painlevé analysis we establish the integrability for a particular form of the nonlinear coefficient. The corresponding soliton solution is shown to be of the self-similar kind. We discuss the implications of the result to the dynamics of attractive Bose–Einstein condensates under Feshbach-managed nonlinearity and explore the possibility of a managed self-similar evolution in 1D condensates.

An adiabatic transport of Bose–Einstein condensates in double-well traps

A complete adiabatic transport of Bose–Einstein condensate (BEC) in a double-well trap is investigated within the mean field approximation. The transfer is driven by the time-dependent (Gaussian) coupling between the wells and their relative detuning. The protocol successfully works in a wide range of both repulsive and attractive BEC interaction. The nonlinear effects caused by the interaction can be turned from detrimental into favourable for the transport. The results are compared with familiar Landau–Zener scenarios using the constant coupling. It is shown that the pulsed Gaussian coupling provides a new transport regime where coupling edges are decisive and convenient switch of the transport is possible.

Scattering of an attractive Bose-Einstein condensate from a barrier: Formation of quantum superposition states

Scattering in one dimension of an attractive ultracold bosonic cloud from a barrier can lead to the formation of two nonoverlapping clouds. Once formed, the clouds travel with constant velocity, in general different in magnitude from that of the incoming cloud, and do not disperse. The phenomenon and its mechanism---transformation of kinetic energy to internal energy of the scattered cloud---are obtained by solving the time-dependent many-boson Schr\"odinger equation. The analysis of the wave function shows that the object formed corresponds to a quantum superposition state of two distinct wave packets traveling through real space.

Zero-temperature Properties of Attractive Bose-Einstein Condensate by Correlated Many-body Approach

We employ a correlated many-body approach to study Bose-Einstein condensation of a magnetically trapped gas of 7Li atoms. The properties of a trapped gas are strongly influenced by the attractive interactions and two-body correlations. The correlations lower the interaction energy. The lowlying collective frequencies have also been calculated. In addition we explore the frequency of surface modes as a function of angular momentum.

The effect of the negative coupling parameter on the spectrum of a trapped Bose gas

The interest in attractive Bose–Einstein Condensates arises due to the chemical instabilities generate when the number of trapped atoms is above a critical number. In this case, recombination process promotes the collapse of the cloud. This behavior is normally geometry dependent. Within the context of the mean field approximation, the system is described by the Gross–Pitaevskii equation. We have considered the attractive Bose–Einstein condensate, confined in a nonspherical trap, investigating numerically and analytically the solutions, using controlled perturbation and self-similar approximation methods. This approximation is valid in all interval of the negative coupling parameter allowing interpolation between weak-coupling and strong-coupling limits. When using the self-similar approximation methods, accurate analytical formulas were derived. These obtained expressions are discussed for several different traps and may contribute to the understanding of experimental observations.

Relation between chaos probability and zero-point number of the Melnikov function for a Bose-Einstein condensate

Abstract We investigate an attractive Bose-Einstein condensate perturbed by a weak traveling optical superlattice. It is demonstrated that under a stochastic initial set and in a given parameter region solitonic chaos appears with a certain probability that is tightly related to the zero-point number of the Melnikov function; the latter depends on the potential parameters. Effects of the lattice depths and wave vectors on the chaos probability are studied analytically and numerically, and different chaotic regions of the parameter space are found. The results suggest a feasible method for strengthening or weakening chaos by modulating the potential parameters experimentally.

Solitons and solitary vortices in pancake-shaped Bose-Einstein condensates

We study fundamental and vortical solitons in disk-morphed Bose-Einstein condensates (BECs) subject to strong confinement along the axial direction. Starting from the three-dimensional (3D) Gross-Pitaevskii equation (GPE), we proceed to an effective 2D nonpolynomial Schroeodinger equation (NPSE) derived by means of the integration over the axial coordinate. Results produced by the latter equation are in very good agreement with those obtained from the full 3D GPE, including cases when the formal 2D equation with the cubic nonlinearity is unreliable. The 2D NPSE is used to predict density profiles and dynamical stability of repulsive and attractive BECs with zero and finite topological charge in various planar trapping configurations, including the axisymmetric harmonic confinement and 1D periodic potential. In particular, we find a stable dynamical regime that was not reported before, viz., periodic splitting and recombination of trapped vortices with topological charge 2 or 3 in the self-attractive BEC.

Angular momentum mixing dynamics of attractive Bose–Einstein condensates on a deformed ring

The Hamiltonian of an attractive N-boson system confined on a ring is diagonalized. The particles have strong correlation as a result of a quantum phase transition in the strong-interaction phase, where the yrast states are quasi-degenerate. It is found that a small deformation of the ring will mix the quasi-degenerate yrast states to form cluster states which break the spatial symmetry explicitly. The time evolution of the average angular momentum in an elliptic ring exhibits strong oscillation.

Remarks on the blow-up rate for critical nonlinear Schrödinger equation with harmonic potential

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a harmonic potential, which models the attractive Bose–Einstein condensate. We establish the sharp lower and upper bounds of blow-up rate as t → T (blow-up time), which improve the result of [Q. Liu, Y. Zhou, J. Zhang, Upper and lower bound of the blow-up rate for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput. 172 (2006) 1121–1132].

Discrete chaotic states of a Bose-Einstein condensate

We examine spatial chaos in a one-dimensional attractive Bose-Einstein condensate interacting with a Gaussian-like laser barrier and perturbed by a weak optical lattice. For a low laser barrier, chaotic regions of the parameters are demonstrated and the chaotic and regular states are illustrated numerically. In the high-barrier case, bounded perturbed solutions that describe a set of discrete chaotic states are constructed for discrete barrier heights and magic numbers of condensed atoms. Chaotic density profiles are exhibited numerically for the lowest quantum number, and analytically bounded but numerically unbounded Gaussian-like configurations are confirmed. It is shown that the chaotic wave packets can be controlled experimentally by adjusting the laser barrier potential.

Localized matter-wave patterns with attractive interaction in rotating potentials

We consider a two-dimensional (2D) model of a rotating attractive Bose-Einstein condensate (BEC), trapped in an external potential. First, a harmonic potential with the critical strength is considered, which generates quasisolitons at the lowest Landau level (LLL). We describe a family of the LLL quasisolitons using both numerical method and a variational approximation (VA), which are in good agreement with each other. We demonstrate that kicking the LLL mode or applying a ramp potential sets it in the Larmor (cyclotron) motion that can also be accurately modeled by the VA. Collisions between two such moving modes may be elastic or inelastic depending on their total norm. If an additional confining potential is applied along with the ramp, it creates a stationary edge state. Applying a kick to the edge state in the direction of the ramp gives rise to a skipping motion in the perpendicular direction. These regimes may be interpreted as the Hall effect for the quasisolitons. Next, we consider the condensate trapped in an axisymmetric quartic potential. Three species of localized states and their stability regions are identified, viz., vortices with arbitrary topological charge $m$, ``crescents'' (mixed-vorticity states), and strongly localized center-of-mass (c.m.) states, alias quasisolitons, shifted off the rotation pivot. These results are similar to those reported before for the model with a combined quadratic-quartic trap. Stable pairs of c.m. states set at diametrically opposite points are found, too. We present a VA which provides for an accurate description of vortices with all values of $m$, and of the c.m. states. We also demonstrate that kicking them in the azimuthal direction sets the quasisolitons in epitrochoidal motion (which is also accurately predicted by the VA), collisions between them being elastic.

Fock-space WKB method for the boson Josephson model describing a Bose-Einstein condensate trapped in a double-well potential

We develop a Fock-space WKB method for a Bose-Einstein condensate (BEC) in a double-well trap, using an analogy with a single particle of either positive or negative mass in a quantum potential. The usual mean-field approach is the classical limit, and the inverse number of atoms plays the role of Planck constant. The ground state of positive-mass particles corresponds to the mean-field fixed point of lower energy, while that of negative mass to the excited fixed point. In the case of attractive BECs above the threshold for symmetry breaking, the ground state is the Schr\"odinger cat state and we relate this to the double-well shape of the potential in Fock space. In the repulsive case, we identify the quantum states corresponding to the mean-field macroscopic self-trapping phenomenon. In particular, the phase-locked ($\ensuremath{\pi}$ phase) macroscopic quantum self-trapping of BECs is related to the double-well shape of the potential in Fock space. The running-phase macroscopic quantum self-trapping state is found to be subject to quantum collapses and revivals. The phase dispersion grows exponentially, reaching the absolute maximum just before the first collapse of the running phase, which may explain the growth of the phase fluctuations seen in the experiment on macroscopic quantum self-trapping.

Bright-matter solitons in a uniformly feeded Bose–Einstein condensate with expulsive harmonic potential

The Gross–Pitaevskii equation, which describes the dynamics of a one-dimensional uniformly feeded attractive Bose–Einstein condensate in an expulsive potential of arbitrary harmonic shape −a2x2+a1x, is solved analytically following the inverse scattering transform method. Within this approach, bright-matter waves are obtained as exact envelope-soliton solutions of the nonlinear Schrödinger equation with a complex harmonic potential. The envelope shapes mimic double-lump pulses of unequal amplitudes symmetric with respect to the potential maximum, moving simultaneously at nonconstant accelerations with amplitudes that vary in time.

Tunable rotary orbits of matter-wave nonlinear modes in attractive Bose–Einstein condensates

We demonstrate that by spatially modulating the Bessel optical lattice where a Bose–Einstein condensate is loaded, we get tunable rotary orbits of nonlinear lattice modes. We show that the radially expanding or shrinking Bessel lattice can drag the nonlinear localized modes to orbits of either larger or smaller radii and the rotary velocity of nonlinear modes can be changed accordingly. The localized modes can even be transferred to the Bessel lattice core when the localized modes' rotations are stopped. Effects beyond the quasi-particle approximation such as destruction of the nonlinear modes by nonadiabatic dragging are also explored.

Collapse times for attractive Bose–Einstein condensates

We argue that the main mechanism for condensate collapse in an attractive Bose–Einstein condensate is the loss of coherence between atoms a finite distance apart, rather than the growth of the occupation number in non-condensate modes. Since the former mechanism is faster than the latter by a factor of approximately 3/2, this helps to dispel the apparent failure of field theoretical models in predicting the collapse time of the condensate.

Dynamical instability and dispersion management of an attractive condensate in an optical lattice

We investigate the stability of an attractive Bose-Einstein condensate in a one-dimensional lattice in the presence of radial confinement. We find that the system is dynamically unstable for low quasimomenta and becomes stable near the band edge, in a specular fashion with respect to the repulsive case. For low interactions the instability occurs via long-wavelength excitations that produce an oscillating density pattern in both real and momentum space instead of spoiling the condensate coherence. This behavior is illustrated by simulations for the expansion of the condensate in a moving lattice.

Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates

We investigate bright matter-wave solitons in the presence of a spatially varying nonlinearity. It is demonstrated that a translation mode is excited due to the spatial inhomogeneity and its frequency is derived analytically and also studied numerically. Both cases of purely one-dimensional and ``cigar-shaped'' condensates are studied by means of different mean-field models, and the oscillation frequencies of the pertinent solitons are found and compared with the results obtained by the linear stability analysis.Numerical results are shown to be in very good agreement with the corresponding analytical predictions.

Bright solitary waves and trapped solutions in Bose–Einstein condensates with attractive interactions

We analyse the static solutions of attractive Bose–Einstein condensates under transverse confinement, both with and without axial confinement. By full numerical solution of the Gross–Pitaevskii equation and variational methods we map out the condensate solutions, their energetic properties and their critical points for instability. With no axial confinement a bright solitary wave solution will tend to decay by dispersion unless the interaction energy is close to the critical value for collapse. In contrast, with axial confinement the only decay mechanism is collapse. The energetic stability of a bright solitary wave solution against decay increases with higher radial confinement. Finally, we consider the stability of dynamical states containing up to four solitons and find good agreement with recent experiments.

Energy criterion of global existence for supercritical nonlinear Schrödinger equation with harmonic potential

This paper is concerned with the supercritical nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose-Einstein condensate under a magnetic trap. We establish two types of new invariant evolution flows. Then, in terms of the Hamiltonian invariants, we derive a new sharp energy criterion for global existence and blowing up of solutions of the equation, which can be precisely computed in the Bose-Einstein condensate.

On a class of quasilinear schrödinger equations

A type of quasilinear Schrodinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sufficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.