Noise effect on the nonadiabatic transition and correction to the tunneling energy gap estimated by the Landau-Zener-Stuckelberg formula

We study effect of noise on the tunneling process of the magnetization in a sweeping field from the viewpoint of the nonadiabatic transition. The tunneling probability depends on the properties of the noise, e.g. the direction, the amplitude, and the relaxation time of the autocorrelation function. We investigate the tunneling process in the presence of noise by solving the time-dependent Schr$\rm{\ddot{o}}$dinger equation numerically, and analyze its parameter dependence. We estimate the effective tunneling gap making use of the Landau-Zener-St$\rm{\ddot{u}}$ckelberg (LZS) formula. The dependence of the gap on the sweeping velocity reproduces the corresponding experimental data qualitatively. For fast sweeping field the estimated gap has no velocity dependence and this gap seems to give the true gap. We, however, show that a correction for the estimated energy gap is necessary and we propose an correction formula.