Model reduction of cavity nonlinear optics for photonic logic: a quasi-principal components approach

Kerr nonlinear cavities displaying optical thresholding have been proposed for the realization of ultra-low power photonic logic gates. In the ultra-low photon number regime, corresponding to energy levels in the attojoule scale, quantum input-output models become important to study the effect of unavoidable quantum fluctuations on the performance of such logic gates. However, being a quantum anharmonic oscillator, a Kerr-cavity has an infinite dimensional Hilbert space spanned by the Fock states of the oscillator. This poses a challenge to simulate and analyze photonic logic gates and circuits composed of multiple Kerr nonlinearities. For simulation, the Hilbert of the oscillator is typically truncated to the span of only a finite number of Fock states. This paper develops a quasi-principal components approach to identify important subspaces of a Kerr-cavity Hilbert space and exploits it to construct an approximate reduced model of the Kerr-cavity on a smaller Hilbert space. Using this approach, we find a reduced dimension model with a Hilbert space dimension of 15 that can closely match the magnitudes of the mean transmitted and reflected output fields of a conventional truncated Fock state model of dimension 75, when driven by an input coherent field that switches between two levels. For the same input, the reduced model also closely matches the magnitudes of the mean output fields of Kerr-cavity-based AND and NOT gates and a NAND latch obtained from simulation of the full 75 dimension model.