Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity that is mathematically described by fractal geometry. Here, we extend the reach of fractal concepts in photonics by experimentally demonstrating multifractality of light in arrays of dielectric nanoparticles that are based on fundamental structures of algebraic number theory. Specifically, we engineered novel deterministic photonic platforms based on the aperiodic distributions of primes and irreducible elements in complex quadratic and quaternions rings. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations in deterministic media with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure–property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, enabling the exploration of novel quantum phases and many-body effects.
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 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 multi-scale-invariant vacuum for quantum nanophotonics
applications.
Integrated quantum devices are at the basis of the realisation of scalable, high-performance quantum technology, including quantum computers and quantum communication schemes, where single photons are emitted, guided, manipulated and detected on a chip. Engineered nano-devices enable the efficient confinement of light and, ultimately, the control of the spontaneous emission dynamics of single emitters, which is crucial for cavity quantum electrodynamics experiments and for the de-
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