|−〉 , that arises through the overlap between states |± , m 〉 and |± , n 〉 . Note that Eq. (11) is somewhat similar to the off diagonal T$_{+}$$_{−}$ and P$_{+}$$_{−}$ terms at t = 0. Note that the number and character of the overlapping resonances are functions of the parameters in the total Hamiltonian, e.g., here g$_{r}$ , g$_{nr}$ , g$_{ph}$ , ω , and ω$_{0}$ . For example, in the extreme case where there is no system-environment coupling (i.e. g$_{r}$ = g$_{nr}$ = g$_{ph}$ = 0), the total Hamiltonian is diagonal in the spin-boson bare state basis. The overlap between the spin-boson bare states then vanishes, and in turn, so does the overlap between the states |±〉 . That is, as expected, in absence the system of coupling to the environment, the spin states are not broadened and thus do not overlap. Consider then the relationship between overlapping resonances as reflected in A$_{+}$$_{−}$ and decoherence as embodied in the system purity S . As an example, we examine dynamics where the environment temperature T = 25 mK and choose to vary g$_{r}$ , keeping the other Hamiltonian parameters fixed. Numerical results are obtained by diagonalizing the Hamiltonian matrix in the zeroth order basis of bare states |± , n 〉 . At the temperature of T = 25 mK, a total of 20 oscillator states contribute, allowing us to diagonalize a 40 × 40 dimensional Hamiltonian matrix to obtain the desired E$_{γ}$, | γ 〉 . To assess the decoherence we find, for each value of g$_{r}$ , the maximum possible purity S$_{max}$ in the system at a fixed time t by optimizing the complex coefficients c$_{±}$ . Results are shown in Fig. 2 where both S$_{max}$ and the overlapping resonance measure A$_{+}$$_{−}$ are plotted as a function of g$_{r}$ at t = 0 . 1 ns. The correspondence between the behavior of these two functions is evident, with larger A$_{+}$$_{−}$ correlating well with larger S$_{max}$ . That is, the greater the overlapping resonances contribution, the more resistant is the optimal superposition to decoherence. Note, significantly, that the optimally resistant state need not be, and is often not, an eigenstate of the spin Hamiltonian, but is rather a superposition of | + 〉 and |−〉 eigenstates. Prominent in the graph (and some presented below) is the DFS point g$_{r}$ = g$_{nr}$ , where the A$_{+}$$_{−}$ curve displays a sharp peak, commensurate with S$_{max}$ reaching its maximum value of unity. That is, maximal purity clearly correlates with the maximum A$_{+}$$_{−}$ . The deviation 8 C. Computational Results