Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations

Abstract

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using Fast Fourier Transforms. To split the system into the quantum harmonic oscillator problem and the remaining part allows to get higher accuracies in many cases, but it requires to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build new methods which combine the advantages of using Fourier methods while solving the timedependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).

12 pages of RevTex4-1, 8 figures; substantially revised and extended version

References in corpus (4)

• The Magnus expansion and some of its applications
• Numerical Solution of the Gross-Pitaevskii Equation for Bose-Einstein Condensation
• Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap
• Fourth-Order Algorithms for Solving the Imaginary Time Gross-Pitaevskii Equation in a Rotating Anisotropic Trap

Cited by in corpus (5)

• Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients
• Asymptotic theory of quasiperiodically driven quantum systems
• Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators
• Symplectic integrators for the matrix Hill's equation and its applications to engineering models
• High-order splitting methods for separable non-autonomous parabolic equations