Time-dependent Gross-Pitaevskii Equation Research Articles

Vortex dipole in a trapped two-dimensional Bose-Einstein condensate

We study the conservative dynamics and stationary configurations of a vortex-antivortex pair in a harmonically trapped two-dimensional Bose-Einstein condensate. We establish the conceptual framework for understanding the stationary states and the topological defect trajectories, through considerations of different mechanisms of vortex motion and the bifurcation of solitonlike stationary solutions. Our insights are based on Lagrangian-based variational calculations, numerical solutions of both the time-dependent and time-independent Gross-Pitaevskii equations, and exact solutions for the noninteracting case.

Solitons and vortices in an evolving Bose-Einstein condensate

Spatiotemporal evolution of a confined Bose-Einstein condensate is studied by numerically integrating the time-dependent Gross-Pitaevskii equation. Self-interference between the successively expanding and reflecting nonlinear matter waves results in spiral atomic density profile, which subsequently degenerates into an embedding structure: The inner part preserves memory of the initial states while the outer part forms a sequence of necklacelike rings. The phase plot reveals a series of discrete concentric belts. The large gradients between adjacent belts indicate that the ring density notches are dark solitons. In the dynamical process, a scenario of vortex-antivortex pairs are spontaneously created and annihilated, whereas the total vorticity keeps invariant.

Stability of Bose-Einstein condensates in a double-well potential

We investigate the stability of the first excited state, the so-called “π-state,” of Bose-Einstein condensates in a double-well potential. From the condition of complex excitation energies, we determine the critical barrier height, above which the π-state is dynamically unstable. We find that the critical barrier height decreases monotonically as the number of condensate atoms increases. We also simulate the dynamics of the π-state by solving the time-dependent Gross-Pitaevskii equation. Our simulation results show that the π-state in the dynamically unstable region exhibits distinctively different behavior from that in the dynamically stable region.

Generating ring dark solitons in an evolving Bose-Einstein condensate

The successive dynamical evolution of a Bose-Einstein condensate confined in a cylindrical well is numerically studied in the framework of the time-dependent Gross-Pitaevskii equation. Interference in the nonlinear matter wave leads to concentric density rings. The phase distribution exhibits a discontinuous sequence of plateaulike belts. Abrupt jumps in the phase between adjacent belts imply large radial superfluid velocity at the borderline. This, however, does not mean large particle current because the corresponding superfluid density is nearly zero. The density zeros along with the large gradient are identified as ring dark solitons, which have a brief lifetime before evolving into other soliton states.

Vortex lattice structures of a Bose-Einstein condensate in a rotating triangular lattice potential

We study the vortex pinning effect in a Bose-Einstein condensate in the presence of a rotating lattice potential by numerically solving the time-dependent Gross-Pitaevskii equation. We consider a triangular lattice potential created by blue-detuned laser beams. By rotating the lattice potential, we observe a structural phase transition from the Abrikosov vortex lattice to the pinned vortex lattice. We investigate the transition of the vortex lattice structure by changing conditions such as angular velocity, strength, and lattice constant of a rotating lattice potential. Our simulation results clearly show that the lattice potential has a strong vortex pinning effect when the vortex density coincides with the density of the pinning points.

Exact solutions of the Gross-Pitaevskii equation in periodic potential in the presence of external source

Exact periodic solutions, solitonlike solutions, singular solitary, and singular trigonometric wave solutions of the time-dependent Gross-Pitaevskii equation (GPE) with elliptic function potential in the presence of external source are analyzed. A simple approach that applies equally to both attractive and repulsive time-dependent GPE and allows one to find an extensive list of explicit periodic solutions of the GPE in terms of the Jacobian elliptic functions is developed. In the limit as the elliptic modulus tends to unity or to zero, the linear solutions, in either the Jacobian elliptic cosine or the Jacobian elliptic function of third order, give solitonlike solutions, while the rational solutions in these elliptic functions lead to singular solitary or trigonometric wave solutions. The stability of these solutions is investigated numerically.

Exact solitonic solutions of the Gross-Pitaevskii equation with a linear potential

We derive classes of exact solitonic solutions of the time-dependent Gross-Pitaevskii equation with repulsive and attractive interatomic interactions. The solutions correspond to a string of bright solitons with phase difference between adjacent solitons equal to pi. While the relative phase, width, and distance between adjacent solitons turn out to be a constant of the motion, the center of mass of the string moves with a constant acceleration arising from the inhomogeneity of the background.

Vortex Lattice Structures of a Bose-Einstein Condensate in a Rotating Lattice Potential

We study vortex lattice structures of a trapped Bose-Einstein condensate in a rotating lattice potential by numerically solving the time-dependent Gross-Pitaevskii equation. By rotating the lattice potential, we observe the transition from the Abrikosov vortex lattice to the pinned lattice. We investigate the transition of the vortex lattice structure by changing conditions such as angular velocity, intensity, and lattice constant of the rotating lattice potential.

Rigorous Derivation of the Gross-Pitaevskii Equation

The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schrödinger equation with a short-scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short-scale correlation structure in the condensate.

Time-dependent multi-orbital mean-field for fragmented Bose–Einstein condensates

The evolution of Bose-Einstein condensates is usually described by the famous time-dependent Gross-Pitaevskii equation, which assumes all bosons to reside in a single time-dependent orbital. In the present work we address the evolution of fragmented condensates, for which two (or more) orbitals are occupied, and derive a corresponding time-dependent multi-orbital mean-field theory. We call our theory TDMF($n$), where $n$ stands for the number of evolving fragments. Working equations for a general two-body interaction between the bosons are explicitly presented along with an illustrative numerical example.

Numerical solution for the Gross–Pitaevskii equation

We solve the time-independent Gross-Pitaevskii (GP) equation which describes the dilute Bose-condensed atoms in harmonic trap at zero temperature by symplectic shooting method (SSM). Both the repulsive nonlinearity and the attractive nonlinearity cases are studied, and the bound state eigenvalues as well as the corresponding wavefunctions are evaluated. We also present the numerical results by studying the time-dependent GP equation, and comparisons are made between the results obtained by the time-independent approach and the time-dependent approach.

Mean-field effects may mimic number squeezing in Bose-Einstein condensates in optical lattices

This paper presents a mean-field numerical analysis, using the full three-dimensional time-dependent Gross-Pitaevskii equation (GPE), of an experiment carried out by Orzel et al. [Science 291, 2386 (2001)] intended to show number squeezing in a gaseous Bose-Einstein condensate in an optical lattice. The motivation for the present work is to elucidate the role of mean-field effects in understanding the experimental results of this work and those of related experiments. We show that the non-adiabatic loading of atoms into optical lattices reproduces many of the main results of the Orzel et al. experiment, including both loss of interference patterns as laser intensity is increased and their regeneration when intensities are lowered. The non-adiabaticity found in the GPE simulations manifests itself primarily in a coupling between the transverse and longitudinal dynamics, indicating that one-dimensional approximations are inadequate to model the experiment.

Dipole oscillations in a Bose-Fermi mixture in the time-dependent approach

We study the dipole collective oscillations in a Bose-Fermi mixture, which are formulated with the time-dependent Gross-Pitaevskii equation and the Vlasov equation, using a dynamical time-dependent approach. We find a large difference in the behavior of fermion oscillations between the time-dependent approach and the usual approaches such as the random-phase approximation. The dipole oscillation of the Bose-Fermi mixture cannot be described with simple center-of-mass motions of the two gases.

Symbolic calculation in development of algorithms: split-step methods for the Gross–Pitaevskii equation

We perform a systematic study of the accuracy of split-step Fourier transform methods for the time dependent Gross-Pitaevskii equation using symbolic calculation. Provided the most recent approximation for the wave function is always used in the nonlinear atom-atom interaction potential energy, every split-step algorithm we have tried has the same-order time stepping error for the Gross-Pitaevskii equation and the Schroedinger equation.

Gross-Pitaevskii equation for Bose particles in a double-well potential: Two-mode models and beyond

There have been many discussions of two-mode models for Bose condensates in a double well potential, but few cases in which parameters for these models have been calculated for realistic situations. Recent experiments lead us to use the Gross-Pitaevskii equation to obtain optimum two-mode parameters. We find that by using the lowest symmetric and antisymmetric wavefunctions, it is possible to derive equations for a more exact two-mode model that provides for a variable tunneling rate depending on the instantaneous values of the number of atoms and phase differences. Especially for larger values of the nonlinear interaction term and larger barrier heights, results from this model produce better agreement with numerical solutions of the time-dependent Gross-Pitaevskii equation in 1D and 3D, as compared with previous models with constant tunneling, and better agreement with experimental results for the tunneling oscillation frequency [Albiez et al., cond-mat/0411757]. We also show how this approach can be used to obtain modified equations for a second quantized version of the Bose double well problem.

Josephson oscillation and induced collapse in an attractive Bose-Einstein condensate

Using the axially symmetric time-dependent Gross-Pitaevskii equation we study the Josephson oscillation of an attractive Bose-Einstein condensate (BEC) in a one-dimensional periodic optical-lattice potential. We find that the Josephson frequency is virtually independent of the number of atoms in the BEC and of the interatomic interaction (attractive or repulsive). We study the dependence of the Josephson frequency on the laser wave length and the strength of the optical-lattice potential. For a fixed laser wave length (795 nm), the Josephson frequency decreases with increasing strength as found in the experiment of Cataliotti et al. [Science 293, 843 (2001)]. For a fixed strength, the Josephson frequency remains essentially unchanged for a reasonable variation of laser wave length around 800 nm. However, the Josephson oscillation is disrupted with the increase of laser wave length beyond 2000 nm leading to a collapse of a sufficiently attractive BEC. These features of a Josephson oscillation can be tested experimentally with present setups.

Monopole oscillations and dampings in a boson and fermion mixture in the time-dependent Gross-Pitaevskii and Vlasov equations

We construct a dynamical model for the time evolution of the boson-fermion coexistence system. The dynamics of bosons and fermions are formulated with the time-dependent Gross-Pitaevsky equation and the Vlasov equation. We thus study the monopole oscillation in the bose-fermi mixture. We find that large damping exists for fermion oscillations in the mixed system even at zero temperature.

Vortex dynamics near the surface of a Bose-Einstein condensate

The center-of-mass dynamics of a vortex in the surface region of a Bose-Einstein condensate is investigated both analytically using a variational calculation and numerically by solving the time-dependent Gross-Pitaevskii equation. We find, in agreement with previous works, that away from the Thomas-Fermi surface, the vortex moves parallel to the surface of the condensate with a constant velocity. We obtain an expression for this velocity in terms of the distance of the vortex core from the Thomas-Fermi surface that fits accurately with the numerical results. We find also that, coupled to its motion parallel to the surface, the vortex oscillates along the direction normal to the surface around a minimum point of an effective potential.

Evolution of a collapsing and exploding Bose-Einstein condensate in different trap symmetries

Based on the time-dependent Gross-Pitaevskii equation we study the evolution of a collapsing and exploding Bose-Einstein condensate in different trap symmetries to see the effect of confinement on collapse and subsequent explosion, which can be verified in future experiments. We make a prediction for the evolution of the shape of the condensate and the number of atoms in it for different trap symmetries (cigar to pancake) as well as in the presence of an optical lattice potential. We also make a prediction for the jet formation in different cases when the collapse is suddenly terminated by changing the scattering length to zero via a Feshbach resonance. In addition to the usual global collapse to the center of the condensate, in the presence of an optical-lattice potential one could also have in certain cases independent collapse of parts of the condensate to local centers, which could be verified in experiments.

Stationary vortex clusters in nonrotating Bose-Einstein condensates

We investigate the recently found stationary vortex cluster states in dilute atomic Bose-Einstein condensates confined by a nonrotating trap, and also present a stationary three-vortex cluster. We find the stationary states by minimizing directly an error norm for the stationary Gross-Pitaevskii equation, and study the dynamic and energetic stability of the resulting states by solving the corresponding Bogoliubov equations for the elementary excitations. The results are verified by integrating the time-dependent Gross-Pitaevskii equation. Contrary to previously reported results, the stationary states were observed to be both energetically and dynamically unstable. The dynamical decay rate of the clusters is typically very slow, but it should be experimentally observable. The most promising circumstances to experimentally generate and observe these structures and their dynamics is in weakly dissipative condensate systems, using phase-imprinting techniques.