Towards Schr\"odinger Cat States in the Second Harmonic Generation

TL;DR

This paper demonstrates that second-harmonic generation can produce macroscopic nonclassical states of light, resembling Schr"odinger cat states, through nonlinear frequency conversion and phase instability.

Contribution

It provides a quantum and classical analysis showing how pump depletion leads to nonclassical superposition states in the pump field during second-harmonic generation.

Findings

Pump develops a Schr"odinger cat-like state from a coherent initial state.

Nonclassical states with negative Wigner function zones are produced.

Classical trajectory methods replicate quantum evolution for large photon numbers.

Abstract

We investigate the quantum evolution of the pump field in second-harmonic generation under strong pump depletion. Starting from a coherent state, the pump develops a nonclassical phase-space structure resembling a Schr\"odinger cat state. This behavior originates from phase instability induced by vacuum fluctuations of the harmonic mode. A rigorous quantum analysis has been performed for mean photon numbers up to $\langle \hat n \rangle = 100$ in pump mode. For larger photon numbers, up to $\langle \hat n \rangle = 10^{7}$, the dynamics have been analyzed using a classical trajectory method with sampled initial conditions that reproduces the main features of the quantum evolution. The results indicate that nonlinear frequency conversion can generate macroscopic superposition-like states of the pump field. Although the resulting state is not pure due to correlations with the second-harmonic wave, it remains non-classical with negative zones of Wigner function. These results indicate that strongly nonlinear frequency conversion can provide a scalable route toward macroscopic nonclassical states of light.