  Single photon-level quantum frequency conversion has recently been
demonstrated using silicon nitride microring resonators. The resonance
enhancement offered by such systems enables high-efficiency translation of
quantum states of light across wide frequency ranges at sub-watt pump powers.
Using a quantum-mechanical Hamiltonian formalism, we present a detailed
theoretical analysis of the conversion dynamics in these systems, and show that
they are capable of converting single- and multi-photon quantum states.
Analytic formulas for the conversion efficiency, spectral conversion
probability density, and pump power requirements are derived which are in good
agreement with previous theoretical and experimental results. We show that with
only modest improvement to the state of the art, efficiencies exceeding 95% are
achievable using less than 100 mW of pump power. At the critical driving
strength that yields maximum conversion efficiency, the spectral conversion
probability density is shown to exhibit a flat-topped peak, indicating a range
of insensitivity to the spectrum of a single photon input. Two alternate
theoretical approaches are presented to study the conversion dynamics: a
dressed mode approach that yields a better intuitive picture of the conversion
process, and a study of the temporal dynamics of the participating modes in the
resonator, which uncovers a regime of Rabi-like coherent oscillations of single
photons between two different frequency modes. This oscillatory regime arises
from the strong coupling of distinct frequency modes mediated by coherent
pumps.
