The first three of these terms depend solely on the magnitude of the c$_{±}$ coefficients, whereas the remainder of the terms depend upon the phases of the coefficients as well. The latter are clearly contributions to S that rely upon quantum interference between the spin states. Note, for comparison later below, that S = 1 denotes a pure state. If the state is completely decohered in the spin basis then ( ρ$_{s}$ )$_{kl}$ = 1 $_{2}$δ$_{kl}$ so that the purity, by Eq. (7), is S = 2(1 / 2) 2 = 1 / 2. With the P , Q , R , T matrices known, one can optimize S with respect to c$_{+}$ and c$_{−}$ to determine the state with maximum purity S at a fixed time. In particular, the relative phase of c$_{+}$ and c$_{−}$ is important if either Q$_{kl}$ or R$_{kl}$ is non-zero, and P$_{kl}$ is non-zero . Although not evident from Eq. (8), numerical results below clearly demonstrate a reliance on this relative phase to maximize S . As noted above, the bare states |± , n 〉 are not eigenstates of H , and will evolve in the presence of the system-environment coupling. Insight into the nature of this time evolution is afforded by expanding the zeroth order state in the exact eigenstates | γ 〉 . For example, a bare state such as | + , j 〉 expands as and evolves as The E$_{γ}$ dependence of the square of the expansion coefficient, i.e. | D$_{γ}$ | 2 ≡ |〈 γ | + , j 〉| 2 provides the energy width over which the zeroth order state | + , j 〉 is spread due to the system-environment interaction. In the simplest cases the inverse of this width provides a qualitative measure of the time scale for the evolution of | + , j 〉 . Hence, the zero order states are indeed resonances with a characteristic width | D$_{γ}$ | $^{2}$. This is the analog, in a bound state spectrum, of the well known resonance in the continuum. Similarly, by analogy to the continuum case, we can define overlapping resonances , as resonances that overlap in energy space, i.e. resonances that share a common | γ 〉 in their 6 | + , j 〉 t = ∑ γ | γ 〉〈 γ | + , j 〉 e − iE$_{γ}$t/ ℏ (10) | + , j 〉 = ∑ γ | γ 〉〈 γ | + , j 〉 , (9) A. Overlapping Resonances IV. OVERLAPPING RESONANCES AND SYSTEM PURITY