  In this paper we propose a novel approach to enhance the efficiency of the
two-photon spontaneous emission process that is driven by the multifractal
optical mode density of photonic structures based on the aperiodic
distributions of Eisenstein and Gaussian primes. In particular, using the
accurate Mie-Lorenz multipolar theory in combination with multifractal
detrended fluctuation analysis, we compute the local density of states of
periodic and aperiodic systems and demonstrate the formation of complete
bandgaps with distinctive fractal scaling behavior for scattering arrays of
dielectric nanocylinders. Moreover, we systematically study the Purcell
enhancement and the most localized optical mode resonances in these novel
aperiodic photonic systems and compute their two-photon spontaneous emission
rates based on the general Green's tensor approach. Our results demonstrate
that the excitation of the highly-resonant critical states of Eisenstein and
Gaussian photonic arrays across broadband multifractal spectra gives rise to
significantly enhanced emission rates compared to what is possible at the
band-edges of periodic structures with comparable size. Besides defining a
novel approach for enhanced quantum two-photon sources on the chip, the
engineering of aperiodic bandgap structures with multifractal mode density may
provide access to novel electromagnetic resonant phenomena in a
multiscale-invariant vacuum for quantum nanophotonics applications.
