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

Journal of Computational Physics 187:1 (2003) 318-342

Authors:

W Bao, D Jaksch, PA Markowich

Abstract:

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. © 2003 Elsevier Science B.V. All rights reserved.
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An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity

SIAM Journal on Numerical Analysis 41:4 (2003) 1406-1426

Authors:

W Bao, D Jaksch

Abstract:

This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrödinger equations (NLSs). The method is explicit, unconditionally stable, and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing NLSs in two dimensions with a linear, cubic, or quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter δ is larger than a threshold value δth. We note that our method can also be applied to solve the three-dimensional Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC).
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Many-particle entanglement in two-component Bose-Einstein condensates

Physical Review A - Atomic, Molecular, and Optical Physics 67:1 (2003)

Authors:

A Micheli, D Jaksch, JI Cirac, P Zoller

Abstract:

The schemes which allow the creation of macroscopically entangled states with a distance 0
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Quantum Information Processing with Quantuim Optics

Annales Henri Poincare 4:SUPPL. 2 (2003)

Authors:

JI Cirac, LM Duan, D Jaksch, P Zoller

Abstract:

We review theoretical proposals for implementation of quantum computing and quantum communication with quantum optical methods.
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Controlling dynamical phases in quantum optics

J OPT B-QUANTUM S O 4:4 (2002) S430-S436

Authors:

T Calarco, D Jaksch, JI Cirac, P Zoller

Abstract:

We review and compare several schemes for inducing precisely controlled quantum phases in quantum optical systems, We focus in particular on conditional dynamical phases, i.e. phases obtained via state- and time-dependent interactions between trapped two-level atoms and ions, We describe different possibilities for the kind of interaction to be exploited, including cold controlled collisions, electrostatic forces, and dipole-dipole interactions.
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