Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods

Abstract

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank-Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned gradient (PG) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement.

Droplet Crystal Ground States of a Dipolar Bose Gas

“…see [22]). We solve for stationary states using gradient flow [23] and conjugate gradient optimization [24,25] techniques adapted to the extended GPE and for constrained total atom number N = dx|ψ| 2 . Solutions are accepted as converged when the residual max |L GP ψ − µψ|/ √ N is smaller than 10 −4 .…”

On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations

“…Moreover, we take the mesh size h and initial data ψ 0 as 19) where φ s0 gs is the ground states (gs) of the SFNLSE (3.6) (if d = 1) and/or (3.8) (with m = λ = 0, if d = 2) with potential given by V (x) (3.18) and fractional order s 0 . The ground states can be computed either by the popular gradient flow method [11][12][13]20] or the recently developed conjugated gradient flow approach [8]. Here, we apply the method proposed in [11].…”

Computation of Ground States of the Gross--Pitaevskii Functional via Riemannian Optimization

“…We consider the case of a BEC trapped in a harmonic potential and rotating at low angular velocities: C trap = r 2 /2, C g = 500, C Ω = 0.4. For this case, the Thomas-Fermi (TF) theory [55] offers a good approximation of the atomic density ρ = |u| 2 of the condensate ρ ≈ ρ TF = (µ − C eff trap )/C g + with the effective trapping potential C eff trap given by (12). By imposing D ρ TF = 1, we can derive analytical expressions for the corresponding approximation of the chemical potential µ ∈ R [57].…”