  Single-photon and correlated two-photon sources are important elements for
optical information systems. Nonlinear downconversion light sources are robust
and stable emitters of single photons and entangled photon pairs. However, the
rate of downconverted light emission, dictated by the properties of
low-symmetry nonlinear crystals, is typically very small, leading to
significant constrains in device design and integration. In this paper, we show
that the principles for spontaneous emission control (i.e. Purcell effect) of
isolated emitters in nanoscale structures, such as metamaterials, can be
generalized to describe the enhancement of nonlinear light generation processes
such as parametric down conversion. We develop a novel theoretical framework
for quantum nonlinear emission in a general anisotropic, dispersive and lossy
media. We further find that spontaneous parametric downconversion in media with
hyperbolic dispersion is broadband and phase-mismatch-free. We predict a
1000-fold enhancement of the downconverted emission rate with up to 105 photon
pairs per second in experimentally realistic nanostructures. Our theoretical
formalism and approach to Purcell enhancement of nonlinear optical processes,
provides a framework for description of quantum nonlinear optical phenomena in
complex nanophotonic structures.
