The name is absent

be changed to

Ψ(r) = Y c_n exρ[-¾<∕_z,                      (2.7)

^z-^j         n

.                                                                 n

We can see now the unitary time evolution in eq. 2.6 has turned into an exponential decay.
The eigenenergy governs the decay rate, and so the eigenstate with the lowest energy,
i.e. the ground state of the system, decays slowest. Therefore, if we start from some
trial wavefunction (which could be a rough guess of the final solution), and apply suitable
renormalization of the wavefunction (e.g fixing the norm and/or the chemical potential)
during the imaginary time propagation, the trial wavefunction will tend towards the ground
state ψo.

2.2 Application to Dipolar Bose-Einstein Condensates

2.2.1 Dipolar Condensatesttheoretical formalism

In this section, we demonstrate a simple application of the split-step Fourier scheme, in
which we look for the ground state of a non-rotating scalar dipolar condensates. We start
from the 3D GP equation, which can be written down directly as following:

⁼ ^^y~ ^+v(^r) + ^col≠(^r)∣² + ⅛ f ^dr'~~Γ-ɜ-ɑɑ^- ⅛(r^,)∣² ψ(r),     (2.8)

∂t 2m                    J ∣r - rψ

where c₀ = ⅛πħ²a∣M, c_d = μ₀μ²∕(4π) with μ = 6μ_s for chromium atoms, one can easily
verify that c_d∣co ≈ 3.6∕α(α_β). The trap is assumed to be harmonic and axially symmetric
V = ^rnω².(x² +y² + Λ²z²). Finally, θ is the polar angle of r - r'. Following the standard

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