Decay-rate distributions using the N-ion model

We use a computational model of N ions located at different distances from graphene, z_i (i = 1, ..., N). The positions z_i ≳ 7 nm are obtained by discretization of the density distribution P(z) calculated from the decay curves. The positions z_i ≲ 7 nm are free parameters that we vary to find the distribution of distances, {zi}i=1,...,N, that best reproduces the emission contrast measurements of Fig. 2a. This variational procedure assumes that the density distribution is smooth. The calculations are accomplished by considering that the emission from every ion is proportional to its excited-state population, which in turn is proportional to 1/(F_Pγ_ed + γ_md + γ_nr), where the theoretical decay-enhancement factor F_P(z_i) is computed from the methods of refs. ^28,29 (see above). Here we assume that γ_nr = 10 Hz for all ions (see Supplementary Note ¹). The discrete distribution {zi}i=1,...,N is converted into a continuous distribution, P(z), and vice versa, by integration and discretization, respectively. We typically use N = 50, where the density distribution already nicely converges.