Generation of Schr\"odinger cat-like states via degenerate dual pump spontaneous four-wave mixing in a $\chi^{(3)}$ microring resonator

TL;DR

This paper presents a full quantum theoretical analysis of generating Schr"odinger cat-like states in a $ ext{chi}^{(3)}$ microring resonator via degenerate dual-pump spontaneous four-wave mixing, accounting for quantum pump depletion and cavity losses.

Contribution

It introduces an exact unitary transformation to decouple nonlinear interactions and studies the resulting non-Gaussian states using a comprehensive quantum approach including dissipation.

Findings

Non-Gaussian features are observed in the signal mode with structured Wigner functions.

High fidelity (>0.9) states are achievable even with dissipation.

The pump mode remains Gaussian, while the signal mode exhibits non-Gaussian, entangled states.

Abstract

We theoretically investigate the generation of non-Gaussian quantum states, specifically Schr\"odinger cat-like states (SCLSs), via degenerate dual-pump spontaneous four-wave mixing in a $\chi^{(3)}$-based microring resonator. By introducing a unitary transformation that exactly decouples the self-phase modulation (SPM) and cross-phase modulation (XPM) terms, we reduce the full nonlinear Hamiltonian to an effective three-mode interaction. The resulting dynamics (decoupled and full Hamiltonians) are studied using the Lindblad master equation, accounting for cavity losses. Unlike semiclassical or parametric approximations, our full quantum mechanical approach explicitly includes quantum pump depletion, which enables the emergence and observation of non-Gaussian features. We compute the Wigner function, photon number distributions, quadrature variances, Fano factor, Schmidt number, and fidelity to characterize the generated states. For the non-dissipative case, we find that the signal mode $\hat{b}_3$ or $\hat{a}_3$ exhibits clear non-Gaussian features with a structured Wigner function and even-dominated photon number distribution, characteristic of an even coherent state. In the presence of dissipation ($\gamma_j = 0.2$), the interference fringes become faint, odd photon numbers appear, and the fidelity with the ideal state remains high ($>0.9$), indicating robustness. The pump mode $\hat{b}_1$ or $\hat{a}_1$ remains Gaussian, while both modes display super-Poissonian statistics and entanglement ($>2$). Our results demonstrate that degenerate dual-pump spontaneous four-wave mixing in microring resonators is a promising platform for generating and controlling cat-like states under dissipative conditions.