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Jisuan wuli

Improved Shooting Method for Gross-Pitaevskii Equation:Interference of Three Bose-Einstein Condensates

Xue‐Shen Liu

Published Jan 1, 2011• 0 citations
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Abstract

An improved shooting method is applied to 1D Gross-Pitaevskii equation,which describes Bose-Einstein condensate of neutral atoms in harmonic trapping potential at zero temperature.Eigenvalues of ground state of condensates with different nonlinear coefficient are given.Interference of three condensates after removing the trapping potential is studied with symplectic method.A periodic evolution is shown.Influence of relative phase on interference of condensates is discussed.

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An improved shooting method is applied to 1D Gross-Pitaevskii equation,which describes Bose-Einstein condensate of neutral atoms in harmonic trapping potential at zero temperature.Eigenvalues of ground state of condensates with different nonlinear coefficient are given.Interference of three condensates after removing the trapping potential is studied with symplectic method.A periodic evolution is shown.Influence of relative phase on interference of condensates is discussed.
The ground state wavefunctions of dilute Bose-condensed atoms in a harmonic trap at T=0 are evaluated by a symplectic shooting method.Stability of the wavefunctions is tested,and a stable wavefunction is used as the initial input of the time-dependent Gross-Pitaevskii equation. The dynamic property of stable wavefunctions is numerically examined in two phase spaces when the harmonic potential is altered suddenly.The figures in the two phase spaces are regular even after a long time of interations.For negative nonlinear coefficients,two eigenvalues related to the same negative nonlinear coefficient are calculated,and the stability of the corresponding two wavefunctions is tested.
The symplectic method is applied to solve numerically the 1D time-dependent Gross-Pitaevskii(GP) equation.The interaction of two Bose-Einstein condensates is investigated when the external potential is exist or the external potential is set to zero.The elastic collision will occur when the external potential is exist.If the external potential is set to zero at t=0,the interference is observed between two condensates.While if the external potential is set to zero at t=2,the interaction between two condensates is complex.
The interference between two condensates with repulsive interaction is investigated numerically by solving the one-dimensional time-dependent Gross–Pitaevskii equation. The periodic interference pattern forms in two condensates, which are prepared in a double-well potential consisting of two truncated harmonic wells centered at different positions. Dark solitons are observed when two condensates overlap. Due to the existence of atom–atom interactions, atoms are transferred among the ground state and the excited states, which coincides with the condensate energy change.