Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

Parametric ladder climbing and the quantum saturation of the threshold for the classical parametric autoresonance due to the zero point fluctuations at low temperatures are discussed. The probability for capture into the chirped parametric resonance is found by solving the Schrödinger equation in the energy basis and the associated resonant phase-space dynamics is illustrated via the Wigner distribution. The numerical threshold for capture into the resonance is compared with the classical and quantum theories in different parameter regimes.

Figure 1

The energy of the chirped parametric oscillator in the quantum PLC (a) and the classical PAR (b) regimes versus the slow time $\tau$. In (b), the Schrödinger simulation (lower red line) is compared to the classical solution (upper green line). The inserts are snapshots of the quantum state in the energy basis at given times. The corresponding Wigner distributions are shown in (c) and (d).

Figure 2

The threshold for PAR versus temperature: the classical simulations (blue upward triangle), the theoretical scaling [Eq. (5)] (dashed line), the Schrödinger simulations (red square) showing quantum saturation, and the generalized theory [Eq. (6)] (solid line). Inset: the capture probability $P(ϵ)$ at $T=0.1$ in the quantum simulations.

Figure 3

The threshold, $P_1^{\mathrm{cr}}$ in ($P_1$, $P_2$) parameter space: Schrödinger simulations (green circle), PLC theory $P_{1,\mathrm{cr}}^{\mathrm{PLC}}=0.237$ (dashed red line), the PAR theory [Eq. (10)] (red dashed dotted line), and the classical simulations (blue square). Black solid line separates the classical PAR and the quantum PLC regimes.