283 research outputs found
The Finite Element Method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation
We consider the time-dependent Gross-Pitaevskii equation describing the
dynamics of rotating Bose-Einstein condensates and its discretization with the
finite element method. We analyze a mass conserving Crank-Nicolson-type
discretization and prove corresponding a priori error estimates with respect to
the maximum norm in time and the - and energy-norm in space. The estimates
show that we obtain optimal convergence rates under the assumption of
additional regularity for the solution to the Gross-Pitaevskii equation. We
demonstrate the performance of the method in numerical experiments
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space-Fractional Gross-Pitaevskii Equation
The present work departs from an extended form of the classical multi-dimensional Gross-Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross-Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system. © 2019 Ahmed S. Hendy et al., published by Sciendo 2019
Blow-up profile of rotating 2D focusing Bose gases
We consider the Gross-Pitaevskii equation describing an attractive Bose gas
trapped to a quasi 2D layer by means of a purely harmonic potential, and which
rotates at a fixed speed of rotation . First we study the behavior of
the ground state when the coupling constant approaches , the critical
strength of the cubic nonlinearity for the focusing nonlinear Schr{\"o}dinger
equation. We prove that blow-up always happens at the center of the trap, with
the blow-up profile given by the Gagliardo-Nirenberg solution. In particular,
the blow-up scenario is independent of , to leading order. This
generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014,
vol. 104, p. 141--156) in the non-rotating case. In a second part we consider
the many-particle Hamiltonian for bosons, interacting with a potential
rescaled in the mean-field manner w\int\_{\mathbb{R}^2} w(x) dx = 1\beta < 1/2a\_N \to a\_*N \to \infty$
Uniform -bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation
In this paper, we consider an energy-conserving continuous Galerkin
discretization of the Gross-Pitaevskii equation with a magnetic trapping
potential and a stirring potential for angular momentum rotation. The
discretization is based on finite elements in space and time and allows for
arbitrary polynomial orders. It was first analyzed in [O. Karakashian, C.
Makridakis; SIAM J. Numer. Anal. 36(6):1779-1807, 1999] in the absence of
potential terms and corresponding a priori error estimates were derived in 2D.
In this work we revisit the approach in the generalized setting of the
Gross-Pitaevskii equation with rotation and we prove uniform -bounds
for the corresponding numerical approximations in 2D and 3D without coupling
conditions between the spatial mesh size and the time step size. With this
result at hand, we are in particular able to extend the previous error
estimates to the 3D setting while avoiding artificial CFL conditions
Finite-temperature dynamics of a single vortex in a Bose-Einstein condensate: Equilibrium precession and rotational symmetry breaking
We consider a finite-temperature Bose-Einstein condensate in a
quasi-two-dimensional trap containing a single precessing vortex. We find that
such a configuration arises naturally as an ergodic equilibrium of the
projected Gross-Pitaevskii equation, when constrained to a finite conserved
angular momentum. In an isotropic trapping potential the condensation of the
classical field into an off-axis vortex state breaks the rotational symmetry of
the system. We present a methodology to identify the condensate and the
Goldstone mode associated with the broken rotational symmetry in the
classical-field model. We also examine the variation in vortex trajectories and
thermodynamic parameters of the field as the energy of the microcanonical field
simulation is varied.Comment: 21 pages, 10 figures. v2: Minor changes and corrections to figures
and text. To appear in PR
Quantum Hall states of bosons in rotating anharmonic traps
We study a model of bosons in the lowest Landau level in a rotating trap
where the confinement potential is a sum of a quadratic and a quartic term. The
quartic term improves the stability of the system against centrifugal
deconfinement and allows to consider rotation frequencies beyond the frequency
of the quadratic part. The interactions between particles are modeled by a
Dirac delta potential. We derive rigorously conditions for ground states of the
system to be strongly correlated in the sense that they are confined to the
kernel of the interaction operator, and thus contain the correlations of the
Bose-Laughlin state. Rigorous angular momentum estimates and trial state
arguments indicate a transition from a pure Laughlin state to a state
containing in addition a giant vortex at the center of the trap (Laughlin
quasi-hole). There are also indications of a second transition where the
density changes from a flat profile in a disc or an annulus to a radial
Gaussian confined to a thin annulus
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