This paper proves that the separation convergence toward the uniform distribution abruptly occurs at times around lnpnq{n for the (time-accelerated by 2) Brownian motion on the sphere with a high dimension. The arguments are based on a new and elementary perturbative approach for estimating hitting times in a small noise context. The quantitative estimates thus obtained are applied to the strong stationary times constructed in [1] to deduce the wanted cut-off phenomenon.