unstable against collapse, just like a Bose gas with a negative scattering length a < O. In a
confined system, a metastable state will be achieved because the positive zero-point kinetic
energy opposes the attractive forces tending to compress the gas to lower the interaction
energy and an energy barrier will be created if number of particles is small enough. Due to
the anisotropy of the dipolar interaction, the sign and the magnitude of the effective dipolar
interaction is strongly influenced by the geometry of the trap.
All these facts make the confined dipolar condensates an interesting candidate for the-
oretical investigation, but also complicated, especially for numerical studies. As we shall
see in this chapter, the Gross-Pitaveskii (GP) equation used to describe the dipolar conden-
sate cannot be solved analytically. Some special numerical techniques have to be used in
solving the GP equation to seek the metastable ground state.
Before we move to discuss the details of the numerical scheme that we use for solving
this GP equation, it is necessary to give a brief introduction to some background knowledge.
2.1 Introduction to split-step Fourier method
Let’s first demonstrate the basic idea of the split-step technique by applying it to solving
the basic GP equation as shown in the chapter 1.
2.1.1 Split-step scheme
As we noticed, the basic GP equation is just the sum of purely space-dependent, and purely
momentum-dependent parts. They are diagonal in real-space and momentum-space repre-
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