Stochastic limit approximation for rapidly decaying systems

The stochastic limit approximation method for “rapid” decay is presented, where the damping rate $\gamma$ is comparable to the system frequency $\Omega ,$ i.e., $\gamma \sim \Omega ,$ whereas the usual stochastic limit approximation is applied only to the weak damping situation $\gamma \ll \Omega .$ The key formulas for rapid decay are very similar to those for weak damping, but the dynamics are quite different. From a microscopic Hamiltonian, the spin-boson model, a Bloch equation containing two independent time scales is derived. This is a useful method to extract the minimal dissipative dynamics at high temperature $k_{B}T\gg ħ\Omega$ and the master equations obtained are of the Lindblad form unlike that of Caldeira and Leggett. The validity of the method is confirmed by comparing the master equation derived through this method with the exact one.