Effect of Duffing nonlinearity on squeezing in a classical parametric amplifier in the unstable regime

Abstract

In quantum information processing, control of quantum entanglement is of central importance, particularly for applications in quantum computing. One established method for generating entanglement is squeezing, which exploits the uncertainty principle: fluctuations in one of a pair of conjugate variables are suppressed at the expense of enhanced fluctuations in the other. A representative approach for squeezing is the parametric amplifier. Depending on the parametric excitation amplitude, the system exhibits regimes of stability and instability of the trivial solution. Conventional studies of squeezing have focused on the stable regime, where the squeezing is maximized near the instability threshold. When the parametric excitation amplitude is increased further and the system enters the unstable regime, the system can reach a steady state due to cubic nonlinearity. However, the squeezing properties of this state have not been fully clarified. In this study, as a first step toward a quantum-mechanical treatment, we investigate squeezing in a classical parametric amplifier in the unstable regime. To enable analytical treatment, we approximate the white noise by a narrowband resonant cosine component and neglect stochasticity. The analysis shows that the steady-state amplitude depends on the phase difference between the noise and the parametric excitation, as in the stable regime, indicating a squeezing effect. Moreover, the conditions under which the steady-state amplitude is maximized and minimized with respect to this phase difference are derived. In addition, we analytically determine how the parametric excitation and noise modify the cubic-nonlinearity-induced bifurcation structure. Based on these findings, we propose a squeezing mechanism that exploits the properties of the unstable regime, unlike conventional approaches. Numerical simulations incorporating stochastic noise reveal the emergence of squeezing, although the effect is weaker than in the stable regime. These results suggest that the unstable regime may offer a possible route for utilizing previously unexplored characteristics.