Time-dependent Gross-Pitaevskii Equation Research Articles

Topological Phase Imprinting in BEC in Presence of Gravitational Field

We report behavior of BEC during topological phase imprinting under the influence of a gravitational field. When the gravitational field is perpendicular to the vortex line as in the experiments, the system is no longer axisymmetric and the vortex is not always stable. We solved the time-dependent Gross-Pitaevskii equation numerically and found that a stable vortex is produced by fine-tuning the field reverse time.

3-Dimensional Dynamics of Vortices in a rotating Bose-Einstein Condensate

We numerically integrate the 3-D time-dependent Gross-Pitaevskii equation suggested by Tsubota et al. [Phys.Rev.A, 65, 023603(2002)] to investigate dynamical transitions of the trapped Bose-Einstein condensate (BEC) from an initial non-vortex state to a steady vortex lattice state after the trigger of the rotation. We use the splitting-fourier method as a numerical scheme to perform stable 3-D numerical simulations with high computational efficiency. We present three stages of characteristic 3-D non-steady dynamics of vortices, i.e., the vortex penetration, the stochastic vortex wandering, the vortex lattice vibration, etc.

Nonlinear Resonant Transport of Bose-Einstein Condensates

The coherent flow of a Bose-Einstein condensate through a quantum dot in a magnetic waveguide is studied. By the numerical integration of the time-dependent Gross-Pitaevskii equation in the presence of a source term, we simulate the propagation process of the condensate through a double barrier potential in the waveguide. We find that resonant transport is suppressed in interaction-induced regimes of bistability, where multiple scattering states exist at the same chemical potential and the same incident current. We demonstrate, however, that a temporal control of the external potential can be used to circumvent this limitation and to obtain enhanced transmission near the resonance on experimentally realistic time scales.

Splitting instability of a multiply charged vortex in a Bose-Einstein condensate

We consider the splitting mechanism of a multiply charged vortex into singly charged vortices in a Bose-Einstein condensate confined in a harmonic potential at zero temperature. The Bogoliubov equations support unstable modes with complex eigenfrequencies (CE modes), which cause the splitting instability without the influence of thermal atoms. The investigation of the excitation spectra shows that the negative-energy (NE) mode plays an important role in the appearance of the CE modes. The configuration of vortices in splitting is determined by the angular momentum of the associated NE mode. This structure has also been confirmed by the numerical simulation of the time-dependent Gross-Pitaevskii equation.

Matter-wave interference, Josephson oscillation and its disruption in a Bose-Einstein condensate on an optical lattice

Using the axially-symmetric time-dependent mean-field Gross-Pitaevskii equation we study the Josephson oscillation in a repulsive Bose-Einstein condensate trapped by a harmonic plus an one-dimensional optical-lattice potential to describe the experiments by Cataliotti et al. [Science 293 (2001) 843, New J. Phys. 5 (2003) 71.1]. After a study of the formation of matter-wave interference upon releasing the condensate from the optical trap, we directly investigate the alternating atomic superfluid Josephson current upon displacing the harmonic trap along the optical axis. The Josephson current is found to be disrupted upon displacing the harmonic trap through a distance greater than a critical distance signaling a superfluid to a classical insulator transition in the condensate.

Matter-wave interference, Josephson oscillation and its disruption in a Bose-Einstein condensate on an optical lattice

Using the axially-symmetric time-dependent mean-field Gross-Pitaevskii equation we study the Josephson oscillation in a repulsive Bose-Einstein condensate trapped by a harmonic plus an one-dimensional optical-lattice potential to describe the experiments by Cataliotti et al. [Science 293 (2001) 843, New J. Phys. 5 (2003) 71.1]. After a study of the formation of matter-wave interference upon releasing the condensate from the optical trap, we directly investigate the alternating atomic superfluid Josephson current upon displacing the harmonic trap along the optical axis. The Josephson current is found to be disrupted upon displacing the harmonic trap through a distance greater than a critical distance signaling a superfluid to a classical insulator transition in the condensate.

Collective oscillations of vortex lattices in rotating Bose-Einstein condensates.

The complete low-energy collective-excitation spectrum of vortex lattices is discussed for rotating Bose-Einstein condensates by solving the Bogoliubov-de Gennes equation, yielding, e.g., the Tkachenko mode recently observed at JILA. The totally symmetric subset of these modes includes the transverse shear, common longitudinal, and differential longitudinal modes. We also solve the time-dependent Gross-Pitaevskii equation to simulate the actual JILA experiment, obtaining the Tkachenko mode and identifying a pair of breathing modes. Combining both approaches allows one to unambiguously identify every observed mode.

Time-dependent Gross-Pitaevskii equation for composite bosons as the strong-coupling limit of the fermionic broken-symmetry random-phase approximation

The linear response to a space- and time-dependent external disturbance of a system of dilute condensed composite bosons at zero temperature, as obtained from the linearized version of the time-dependent Gross-Pitaevskii equation, is shown to result also from the strong-coupling limit of the time-dependent BCS (or broken-symmetry RPA) approximation for the constituent fermions subject to the same external disturbance. In this way, it is possible to connect excited-state properties of the bosonic and fermionic systems by placing the Gross-Pitaevskii equation in perspective with the corresponding fermionic approximations

Gray solitons and phonons in repulsive Bose-Einstein condensates confined in one-dimensional nonharmonic traps

We numerically solve the time-dependent Gross-Pitaevskii equation (GPE) that describes the evolution of an elongated dilute repulsive atomic Bose-Einstein condensate trapped in a one-dimensional (1D) nonharmonic potential. We find that the gray solitons, which are propagative solutions of the 1D GPE, traveling at an initial constant velocity, smaller than the speed of sound, oscillate through the trapped condensate, but that this oscillatory motion is accompanied by a spontaneous emission of small sound waves. By examining the gray soliton trajectory and its velocity in the trapped repulsive Bose-Einstein condensate, we show that the oscillatory motion is uniform and nondissipative except at the returning points of the gray soliton, where it exhibits a slight radiative acceleration (antidamping). Our numerical results are in good agreement with previous theoretical predictions, but show the need to take radiation emission into account.

Splitting of a doubly quantized vortex through intertwining in Bose-Einstein condensates

The stability of doubly quantized vortices in dilute Bose-Einstein condensates of 23Na is examined at zero temperature. The eigenmode spectrum of the Bogoliubov equations for a harmonically trapped cigar-shaped condensate is computed and it is found that the doubly quantized vortex is spectrally unstable towards dissection into two singly quantized vortices. By numerically solving the full three-dimensional time-dependent Gross-Pitaevskii equation, it is found that the two singly quantized vortices intertwine before decaying. This work provides an interpretation of recent experiments [A. E. Leanhardt et al. Phys. Rev. Lett. 89, 190403 (2002)].

Mean-field model for Josephson oscillation in a Bose-Einstein condensate on an one-dimensional optical trap

Using the axially-symmetric time-dependent Gross-Pitaevskii equation we study the phase coherence in a repulsive Bose-Einstein condensate (BEC) trapped by a harmonic and an one-dimensional optical lattice potential to describe the experiment by Cataliotti {\it et al.} on atomic Josephson oscillation [Science {\bf 293}, 843 (2001)]. The phase coherence is maintained after the BEC is set into oscillation by a small displacement of the magnetic trap along the optical lattice. The phase coherence in the presence of oscillating neutral current across an array of Josephson junctions manifests in an interference pattern formed upon free expansion of the BEC. The numerical response of the system to a large displacement of the magnetic trap is a classical transition from a coherent superfluid to an insulator regime and a subsequent destruction of the interference pattern in agreement with the more recent experiment by Cataliotti {\it et al.} [e-print cond-mat/0207139].

Bogoliubov spectrum and Bragg spectroscopy of elongatedBose-Einstein condensates

The behaviour of the momentum transferred to a trappedBose-Einstein condensate by a two-photon Bragg pulse reflectsthe structure of the underlying Bogoliubov spectrum. Inelongated condensates, axial phonons with different numbers ofradial nodes give rise to a multibranch spectrum which can beresolved in Bragg spectroscopy, as shown by Steinhauer et al (2003 Phys. Rev. Lett. 90 060404). Here wepresent a detailed theoretical analysis of this process. We calculate the momentum transferred by numerically solving the time-dependent Gross-Pitaevskii (GP) equation. In the case of acylindrical condensate, we compare the results with thoseobtained by linearizing the GP equation andusing a quasiparticle projection method. This analysis showshow the axial-phonon branches affect the momentum transfer, inagreement with our previous interpretation of the observeddata. We also discuss the applicability of this type ofspectroscopy to typical available condensates, as well as the role of nonlinear effects.

Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap

We study the numerical resolution of the time-dependent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used to simulate the dynamics of Bose-Einstein condensates. Considering condensates trapped in harmonic potentials, we present an efficient algorithm by making use of a spectral-Galerkin method, using a basis set of harmonic-oscillator functions, and the Gauss-Hermite quadrature. We apply this algorithm to the simulation of condensate breathing and scissor modes.

Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation

We study the numerical solution of the time-dependent Gross–Pitaevskii equation (GPE) describing a Bose–Einstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d Gross–Pitaevskii equation and obtain a four-parameter model. Identifying ‘extreme parameter regimes’, the model is accessible to analytical perturbation theory, which justifies formal procedures well known in the physical literature: reduction to 2d and 1d GPEs, approximation of ground state solutions of the GPE and geometrical optics approximations. Then we use a time-splitting spectral method to discretize the time-dependent GPE. Again, perturbation theory is used to understand the discretization scheme and to choose the spatial/temporal grid in dependence of the perturbation parameter. Extensive numerical examples in 1d, 2d and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/nonzero initial phase data are presented to demonstrate the power of the numerical method and to discuss the physics of Bose–Einstein condensation.

Dynamics of a collapsing and exploding Bose-Einstein condensed vortex state

Using the time-dependent mean-field Gross-Pitaevskii equation we study the dynamics of small repulsive Bose-Einstein condensed vortex states of ^{85}Rb atoms in a cylindrical trap with low angular momentum hbar L per atom (L <= 6), when the atomic interaction is suddenly turned attractive by manipulating the external magnetic field near a Feshbach resonance. Consequently, the condensate collapses and ejects atoms via explosion and a remnant condensate with a smaller number of atoms emerges that survives for a long time. Detail of this collapse and explosion is compared critically with a similar experiment performed with zero angular momentum (L=0). Suggestion for future experiment with vortex state is made.

Josephson spectroscopy of a dilute Bose-Einstein condensate in a double-well potential

The dynamics of a Bose-Einstein condensate in a double-well potential are analysed in terms of transitions between energy eigenstates. By solving the time-dependent and time-independent Gross-Pitaevskii equation in one dimension, we identify tunnelling resonances associated with level crossings, and determine the critical velocity that characterises the resonance. We test the validity of a non-linear two-state model, and show that for the experimentally interesting case, where the critical velocity is large, the influence of higher-lying states is important.

Mean-field description of collapsing and exploding Bose-Einstein condensates

We perform numerical simulations based on the time-dependent mean-field Gross-Pitaevskii equation to understand some aspects of a recent experiment by Donley et al. [Nature (London) 412, 295 (2001)] on the dynamics of collapsing and exploding Bose-Einstein condensates of ${}^{85}\mathrm{Rb}$ atoms. These authors manipulated the atomic interaction by an external magnetic field via a Feshbach resonance, thus changing the repulsive condensate into an attractive one, and vice versa. In the actual experiment they suddenly changed the scattering length of atomic interaction from a positive to a large negative value on a preformed condensate in an axially symmetric trap. Consequently, the condensate collapsed and ejected atoms via explosion. We find that the present mean-field analysis can explain some aspects of the dynamics of the collapsing and exploding Bose-Einstein condensates.

Bose-Einsteincondensation dynamics from the numerical solution of theGross-Pitaevskii equation

We study certain stationary and time-evolution problems oftrapped Bose-Einstein condensates using the numerical solutionof the Gross-Pitaevskii (GP) equation with both spherical and axialsymmetries. We consider time-evolution problems initiated by suddenlychanging the interatomic scattering length or harmonic trappingpotential in a stationary condensate. These changesintroduce oscillations in the condensate which are studied indetail. We use a time iterative split-step method for thesolution of the time-dependent GP equation, whereall nonlinear and linear non-derivative terms are treatedseparately from the time propagation with the kinetic energyterms. Even for an arbitrarily strong nonlinear term this leadsto extremely accurate and stable results after millions of timeiterations of the original equation.

Free expansion of attractive and repulsive Bose-Einstein condensed vortex states

Free expansion of attractive and repulsive Bose-Einstein condensed vortex states formed in an axially symmetric trap is investigated using the numerical solution of the time-dependent Gross-Pitaevskii equation. In a repulsive condensate the vortex-core radius is much smaller than the radial root-mean-square (rms) radius, which makes the experimental observation of the vortex core difficult. The opposite is found to be true in an attractive condensate that makes it a better candidate for experimental observation. Also, in all cases the ratio of vortex-core radius to radial rms radius increases as the angular momentum of the vortex increases. This makes the vortex states with higher angular momenta more suitable for experimental confirmation.

Collapse of attractive Bose-Einstein condensed vortex states in a cylindrical trap.

The quantized vortex states of a weakly interacting Bose-Einstein condensate of atoms with attractive interatomic interaction in an axially symmetric harmonic oscillator trap are investigated using the numerical solution of the time-dependent Gross-Pitaevskii equation obtained by the semi-implicit Crank-Nicholson method. The collapse of the condensate is studied in the presence of deformed traps with the larger frequency along either the radial or the axial direction. The critical number of atoms for collapse is calculated as a function of the vortex quantum number L. The critical number increases with increasing angular momentum L of the vortex state but tends to saturate for large L.