Theoretical Study of Quantum Coherence in Spin-Boson System

We study a two-state system which is coupled to a bosonic environment with the Ohmic spectral density. In this paper, two methods are used. One is the quantum Langevin equation and the other the reduced density matrix. Through these investigations the time reversal symmetry is emphasized because the spectral density of the environment gives rise to a long-ranged collision effect on the spin part. By using the second method, we find, at T=0, (1) a purely oscillatory motion for alpha < alpha_{c} (alpha is a dimentionless dissipation coefficient and alpha_{c} =~ 0.27); (2) a divergent frequency of the oscillation for alpha = alpha_{c}; (3) a localization for alpha > alpha_{c}. On the other hand, a simple decoupling of the Langevin equation merely gives the lowest order calculation in alpha. It is notable that the approximation used here is reliable regardless of a range of alpha. Concerning this a detailed illustration is shown.