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Physical Review A

Bifurcation and quantum delocalized states in second-harmonic generation

S. T. Gevorkyan

Published Jun 15, 2000• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.62.013813

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Photon number and phase fluctuations and correlations in the second harmonic generation are discussed. The new Pegg-Barnett Hermitian phase formalism is used to deal with the phase properties of the field. The method of numerical diagonalization of the interaction Hamiltonian is applied to find the state evolution and, consequently, the number and phase quantum fluctuations. It is shown that the joint phase probability distribution evolves into a multi-peak structure indicating clearly the transition from the second harmonic to the down-conversion regime, and back. The evolution of all relevant quantities is illustrated graphically, and their dependence on the mean number of photons of the initial field is shown explicitly. The number-phase uncertainty product for the fundamental mode is plotted making evident the abrupt transition from the low to the high level of quantum fluctuations.
We show how quantum instability is displayed in the von Neumann entropy and in the Wigner function. For this purpose, an intracavity second-harmonic generation close to the Hopf bifurcation range is studied. We examine the role of dissipation in unstable dynamics and the formation of the quantum states of the cavity modes, and discuss contrast ensemble behavior with that of individual realization on the basis of a quantum-jump simulation method. Namely, it is found that the Wigner functions for fundamental and second-harmonic modes prepared initially in a vacuum state acquire the three-hump structure due to phase symmetry breaking in the bifurcation range. The time evolution of coherent states leads to long-lived swing of the system between two side humps in a phase space.