Operator differential equation approach to the dissipative two-state system

We consider a two-state system linearly coupled to a collection of harmonic oscillators, the spin-boson Hamiltonian. Expanding the density matrix in terms of Pauli spin matrices, a set of coupled operator differential equations of motion is obtained by solving the Heisenberg equation. Using the cumulant expansion methods, the expectation values of the Pauli spin matrices are obtained and found to be valid for both coherent and incoherent regimes. The results are compared with other approaches