ashtonsbradley / fgpeexamples.jl

simple examples to start using FourierGPE; Gross-Pitaevskii equation; Fourier spectral method

FGPEexamples.jl

Simple examples of using the FourierGPE library for solving the Gross-Piteavskii equation in julia.

Some of the larger media files are not tracked by this repository. To run them locally, first

git clone https://github.com/AshtonSBradley/FGPEexamples.jl

then edit flist in /src/FGPEexamples.jl to choose which examples to run (by default will take ~15 minutes), and which output to generate (html by default, but also .pdf, .ipynb can be enabled).

To run the examples in flist, make sure you are in the package directory and do

using Pkg, FGPEexamples
Pkg.pkg"activate ."
Pkg.pkg"instantiate"
FGPEexamples.weave_all()

The output will be created in the docs directory.

Bright soliton

The bright soliton is an eigenstate of the GPE with attractive interactions.

The bright soliton example thus provides a simple test of periodicity for a Fourier method, and that the dispersion and interactions balance correctly.

Dark Soliton

A dark soliton is a non-trivial solution of the Gross-Piteavskii equation involving a localized density dip, and associated phase jump. A moving dark soliton has depth related to its velocity. When the depth drops to zero, the velocity vanishes.

If we phase and density imprint a dark soliton onto a BEC in a harmonic trap, the soliton position undergoes simple harmonic motion.

In the dark soliton example we find a ground state of a harmonic trap and the imprint a dark soliton in the trap center, observing simple harmonic motion:

Vortex in a 2D harmonic trap

A quantum vortex will precess at a frequency that is known analytically providing a simple test of both GPE evolution and vortex imprinting methods.

In the vortex precession example we find a ground state, imprint a vortex, evolve in real time:

The frequency compares well with the analytical result.

Quench in a 3D periodic box

The same code will generate 3D, 2D, 1D systems (n-D is also available).

Here a 3D quench from random initial conditions exhibits proliferation of vortices at short times giving way to eventual relaxation to the quiescent ground state.

Quench in a 3D harmonic trap

More experimentally relevant, a quench in an oblate 3D parabolic trap nucleates a quantum vortices preferentially aligned with the tight axis that eventually decay due to dissipation by migrating to the outer superfluid boundary. Physically, the dissipation is modelling coupling to a thermal cloud at rest in the laboratory frame. We can also observe the formation of a higher energy vortex aligned along the weak axis that damps out on a much shorter timescale.

Quench in 3D into planar confinement

A 3D quench into planar confinement shows some of the rich dynamics as vortices are nucleated, interact with each other and with phonons, and decay. Download this repository and run the examples to see the Kibble-Zurek mechanism in action!

Quench in 3D into tube confinement

A 3D quench into tube confinement reveals how much more fragile the lower dimensional excitations are. Without the topological stability of vortices, dark solitons are free to decay. The stable remnant of the phase transition in this case is a persistent current.

Jones-Roberts soliton evolving in 2D

The JR-soliton is a quasisolitonic solution of the repulsive GPE. In this example it can be seen propagating without change of shape