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 _«,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 φn
of the GP equation, which are defined by HgpΦ∏ = E,,φn,^ih the eigenenergies En. So the
time evolution under the GP equation thus is:

Ψ(z)= У cπexp[-¾0fl                    (2.6)

z-1          n

n

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



More intriguing information

1. FUTURE TRADE RESEARCH AREAS THAT MATTER TO DEVELOPING COUNTRY POLICYMAKERS
2. Tastes, castes, and culture: The influence of society on preferences
3. EDUCATIONAL ACTIVITIES IN TENNESSEE ON WATER USE AND CONTROL - AGRICULTURAL PHASES
4. Making International Human Rights Protection More Effective: A Rational-Choice Approach to the Effectiveness of Ius Standi Provisions
5. Biological Control of Giant Reed (Arundo donax): Economic Aspects
6. The name is absent
7. Imputing Dairy Producers' Quota Discount Rate Using the Individual Export Milk Program in Quebec
8. Influence of Mucilage Viscosity On The Globule Structure And Stability Of Certain Starch Emulsions
9. Deletion of a mycobacterial gene encoding a reductase leads to an altered cell wall containing β-oxo-mycolic acid analogues, and the accumulation of long-chain ketones related to mycolic acids
10. Towards a Mirror System for the Development of Socially-Mediated Skills