Towards Schrödinger Cat States in the Second Harmonic Generation

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 Schrödinger 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.

Figures (4)

• Figure 1: Dependence of the mean pump wave photon number on dimensionless interaction time $gt$: circles — quantum calculations, red line — calculations by classical simulation method; a — initial photon number 50, b — initial photon number 100. The classical simulation used averaging over 1000 trajectories (samples).
• Figure 2: Wigner quasiprobability for the pump wave: a — initial photon number 50, time $gt =$ 0.463 (Fig. 2a), quadrature probability densities are presented on the sides; b — initial photon number 100, time $gt =$ 0.352 (Fig. 2b), inset shows the central part of the Wigner function.
• Figure 3: Time of first maximum $gt_{\text{max}}$ as a function of the initial mean pump wave photon number (circles) and approximating line according to formula (\ref{['eq:gtmax']}).
• Figure 4: Distribution of resulting pump field amplitudes in the classical picture. Points correspond to individual realizations of solutions with random initial values; a — initial mean photon number $\langle n \rangle = 50$, time $gt_{\text{max}} = 0.463$, b — initial mean photon number $\langle n \rangle = 100$, time $gt_{\text{max}} = 0.352$ (see Fig.\ref{['fig:fig1']}).