The ground state and the spectral structure of lower-lying excited states of a dissipative two-level system coupled to a sub-Ohmic bath ($s=1/2$) with nonzero bias have been studied using the unitary transformation method. By calculating the ground-state entanglement entropy, the ground-state average of $\langle {\sigma }_z\rangle {}_G$, and the static susceptibility of the two-level system, we explore the nature of the transition (crossover) between the delocalized and localized state of the two-level system. Furthermore, we calculate the time-dependent expectation $\langle {\sigma }_z(t)\rangle$ and the time evolution of the entanglement entropy to show that, when the system undergoes a transition (crossover) from the delocalized to the localized state, the time evolution of the two-level system changes from coherent to decoherent dynamics.

• Received 28 February 2011

DOI:https://doi.org/10.1103/PhysRevE.84.011114