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Physical Review A

Quantum trajectories and classical attractors in second-harmonic generation

Xiping Zheng, Craig M. Savage

Published Jan 1, 1995• 17 citations
Publisher

Abstract

We numerically investigate the classical limit of quantum trajectories in optical second-harmonic generation. This is a dissipative system of two nonlinearly coupled harmonic oscillators, corresponding to the fundamental and second-harmonic optical cavity modes. Classically it is described by a system of nonlinear differential equations which have a range of attractors: fixed points, limit cycles, and chaotic attractors. We consider the field-amplitude mean values corresponding to the quantum trajectory wave functions. We find that in the classical limit of large photon number the trajectories of these mean values approach the corresponding classical attractors. This is because the mean values of operator products factorize into products of mean values in the classical limit.

DOI
10.1103/physreva.51.792

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We numerically investigate the classical limit of quantum trajectories in optical second-harmonic generation. This is a dissipative system of two nonlinearly coupled harmonic oscillators, corresponding to the fundamental and second-harmonic optical cavity modes. Classically it is described by a system of nonlinear differential equations which have a range of attractors: fixed points, limit cycles, and chaotic attractors. We consider the field-amplitude mean values corresponding to the quantum trajectory wave functions. We find that in the classical limit of large photon number the trajectories of these mean values approach the corresponding classical attractors. This is because the mean values of operator products factorize into products of mean values in the classical limit.
Summary form only given. The situation when nonlinear interactions involve a large number of photons, N, is quite typical of many of the problems of quantum and nonlinear optics. To describe the dynamics of such quantum systems a classical approach is often used. In this theoretical work we compare classical and quantum dynamics of several dissipative quantum optical systems, which have a limit cycle in the classical limit N/spl rarr//spl infin/. In particular, we consider the dynamics of intracavity second harmonic generation (SHG) described by the Hamiltonian.
Abstract The stochastic Schrödinger equation, a novel method for describing the dissipative dynamics of open quantum systems interacting with their environment, is applied to some nonlinear optical processes, namely nonlinear absorption and second‐harmonic generation, the latter also in the range of instabilities and classical chaotic behaviour. Where possible the results are compared with numerical simulations of the positive P representation, which in some cases are found to give unreliable results.
We develop a semiclassical method to determine the nonlinear dynamics of dissipative quantum optical systems in the limit of large number of photons N; it is based on the 1/N-expansion and the quantum-classical correspondence. The method is used to tackle two problems: the study of the dynamics of nonclassical state generation in higher order anharmonic dissipative oscillators and the establishment of the difference between the quantum and classical dynamics of the second-harmonic generation in a self-pulsing regime. In addressing the first problem, we obtain an explicit time dependence of the squeezing and the Fano factor for an arbitrary degree of anharmonism in the short-time approximation. For the second problem, we analytically find a characteristic time scale at which the quantum dynamics differs insignificantly from the classical one.
Second harmonic generation and subharmonic generation in a cavity are described by the usual master equation (1) for the statistical operator of the system of two modes (fundamental and second harmonic). The master equation is equivalent to a partial differential equation for the Wigner function which is a generalized Fokker-Planck equation involving partial derivatives of the first, second and third order (2). In the semi-classical limit the third order derivatives are negligible and the Wigner distribution satisfies the Fokker-Planck equation equivalent to the Langevin equation with the formally classical Gaussian white noise ξ1, ξ2 with the only non-vanishing correlation coefficients β1, β2 are the mode amplitudes (normalized to photon numbers), g is the coupling constant, Δ1, 2 the frequency mismatch, x is the damping rate, Fp the amplitude of the pump field.
For the process of intracavity second-harmonic generation in the region of unstable dynamics, the joint fluctuations of numbers of photons and phases of the fundamental mode and second harmonic are studied in positive P representation. It is shown that above the bifurcation point of the optical system the distribution functions of the phases of the fundamental mode and second harmonic have a two-component structure. The quantum trajectories of the phases of these modes are not localized in the components of their states, which demonstrates the presence of quantum-mechanical interference between these components. In both components of each state the same values of numbers of photons are realized. It is shown that in the case of coherent initial states of both modes the joint distribution functions of the numbers of photons and phases are in the oscillatory regime. With time, the modes pass from a two-component delocalized state to a one-component one and then back to the two-component state. The time of oscillation of the distribution function is of the order of the photon's lifetime in the cavity volume. Self-matching of phases of interacting modes is investigated above the bifurcation point.
We study the difference between quantum and classical behavior in a pair of nonidentical cavities with second-harmonic generation. In the classical limit, each cavity has a limit-cycle solution, in which the photon number oscillates periodically in time. Coupling between the cavities leads to synchronization of the oscillations and classical correlations between the cavities. In the quantum limit, there are quantum correlations due to entanglement. The quantum correlations persist even when the cavities are far off resonance with each other, in stark contrast with the classical case. We also find that the quantum and classical limits are connected by an intermediate regime of almost no correlations. Our results can be extended to a wide variety of quantum models.