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Canadian Journal of Physics

Nonlinear effects in the squeezing spectrum of a parametric amplifier

Haider R. Zaidi

Published Oct 1, 1988• 7 citations
Publisher

Abstract

A general nonlinear theory is formulated for the calculation of the squeezing spectrum of a parametric amplifier. The lowest order nonlinear corrections are calculated for both modes below the threshold, and the results are compared with linear theory. The corrections to the spectrum are found to be significant.
DOI
10.1139/p88-140

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The squeezing properties of a model of a degenerate parametric amplifier with absorption losses and an added fourth-order nonlinearity have been analyzed. The approach used consists of obtaining the Langevin equation for the optical field from the Heisenberg equation provided that a linearization procedure is valid. The steady states of the deterministic equations have been obtained and their local stability has been analyzed. The stationary covariance matrix has been calculated below and above threshold. Below threshold, a squeezed vacuum state is obtained and the nonlinear effects in the fluctuations have been taken into account by a Gaussian decoupling. In the case above threshold, a phase-squeezed coherent state is obtained and numerical simulations allowed to compute the time interval, depending on the loss parameter, on which the system jumps from one stable state to the other. Finally, the variances numerically determined have been compared with those obtained from the linearized theory and the limits of validity of the linear theory have been analyzed. It has become clear that the nonlinear contribution may perhaps be profitably used for the construction of above-threshold squeezing devices.
We perform discrete-mode calculations for a parametric amplifier with a quantum pump and discuss some of the limitations on calculations of this sort in quantum optics. We calculate corrections to the squeezing due to pump quantum fluctuations to order N${\ifmmode\bar\else\textasciimacron\fi{}}^{\mathrm{\ensuremath{-}}2}$, for a pump initially in a coherent state with average photon number N\ifmmode\bar\else\textasciimacron\fi{}. We find that the limit to the variance of the squeezed quadrature due to the quantum nature of the pump goes as N${\ifmmode\bar\else\textasciimacron\fi{}}^{\mathrm{\ensuremath{-}}1/2}$. .AE
The quantum theory of the degenerate parametric amplifier is usually treated in the parametric approximation where the pump field is treated classically. In this paper we present a fully quantized theory of this nonlinear optical device using a path-integral approach. A perturbation series, the first term of which corresponds to the parametric approximation, is employed to evaluate explicitly the coherent-state propagator. The question of the validity of the parametric approximation is considered and the conditions under which this approximation is justified are elucidated. Finally, certain correlation functions for the signal-mode operators are calculated that are needed to study squeezed states. It is shown that the quantum nature of the pump field tends to decrease the squeezing.
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.