RKFD scheme for quantum reflection model of Bose-Einstein Condensates (BEC) from silicon surface

We applied the numerical combination of Runge-Kutta and Finite Difference (RKFD) scheme for a quantum reflection model of Bose-Einstein condensate (BEC) from a silicon surface. It is by the time-dependent Gross-Pitaevskii equation (GPE), a non-linear Schrödinger equation (NLSE) in the context of quantum mechanics. The role of cut-off potential δ and negative imaginary potential 𝑉𝑖𝑚 is essential to estimating non-interacting BEC reflection models. Relying on these features, we performed a numerical simulation of the BEC quantum reflection model and calculated the effect of reflection probability R versus incident speed 𝑣𝑥. The model is based on the three rapid potential variations: positive-step potential +𝑉𝑠𝑡𝑒𝑝, negative-step potential −𝑉𝑠𝑡𝑒𝑝, and Casimir-Polder potential 𝑉𝐶𝑃. As a result, the RKFD numerical scheme was successfully set up and applied to the quantum reflection model of BEC from the silicon surface. The numerical simulation results show that the reflection probability R decays exponentially to the incident speed 𝑣𝑥.