Bosonization of a two-level system with dissipation

We summarize in this work the relations of the spin-boson Hamiltonian, which was recently studied in connection with the phenomenon of quantum coherence in the presence of dissipation, to three different fermionic Hamiltonians. These relations were obtained through well-known equivalences between Fermi and Bose operators in one dimension. The fermionic Hamiltonians correspond to (a) a two-level system coupled linearly with a fermionic bath, (b) the resonant-level model, and (c) the anisotropic Kondo model. For the first Hamiltonian we reobtain and discuss the relationship between the dimensionless dissipation coefficient α and the phase shift of the fermions. The resonant-level model allows us to study the properties of the two-level system for values of α around (1/2). At α=(1/2) the model reduces to the Toulouse limit, where the Hamiltonian is exactly soluble. A pure exponential decay is obtained for the relaxation of P(t)=〈${\sigma }_z$(t)〉 for t>0, given that for t<0 the system is known to be localized in one of the two states. The comparison with the Kondo model gives the long-time-limit behavior of the system for (1/2)