The name is absent

fact that .2⁹ and ʃ do not commute with each other. The splitting error can be reduced
noticeably by using the Baker-Campbell-Hausdorff formula [4] to obtain the higher order
split-step scheme. For example, in our calculation, we usually adopt the second-order
scheme in our code as following:

Advancing __1Ж«,_Т1 . . FT .≈, .    ∆t. Advancing -,⅛>Λ,._f∙, , ₄

Ψ,-----→ e ^{t,λ г} Ψ,(Z) —> Ψ,(Z + — )-------» e Ψ,(r)

²                                 (2.5)

inverseFT          ΔZ Advancing . ₄∕δ<._t. _z At

-------> Ψ,(Z + —)-------> e * ^x∖!_i(t + —) → Ψ,∙(Z + At)

2.1.2 Solving Gross-Pitaevskii equation in imaginary time

In broad terms, the GPE is a nonlinear Schrodinger equation, for which a lot of numerical
techniques have been developed. For our task to search the ground-state, -a very effective
method available is to evolving the GP equation in imaginary time. Through a Wick ro-
tation τ = it, the Schrodinger equation is changed to a reaction-diffusion equation in real
time. As a result, by advancing the diffusion equation in time until local equilibrium is
reached, the ground state can be obtained. Using the basic GP equation as an example, the
imaginary time propagation method works in the following way:

First let us consider a condensate wave function Ψ as a superposition of eigenstates φ_n
of the GP equation, which are defined by HgpΦ∏ = E,,φ_n,^i∖h the eigenenergies E_n. So the
time evolution under the GP equation thus is:

Ψ(z)= У c_πexp[-¾0_fl                    (2.6)

^z-¹          n

n

where the coefficients c_n are defined by the expansion the initial condition Ψ(z = O) =
∑_π c_nφ_rι. If we propagate the GP equation in imaginary time, the above time evolution will

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