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Physical Review A

Bifurcations and multistability in two-photon processes

Daniel F. Walls, C. V. Kunasz, P. D. Drummond, P. Zoller

Published Jul 1, 1981• 22 citations
Publisher

Abstract

We consider an intracavity nonlinear medium interacting with two modes of the electromagnetic field via a two-photon process. The cavity is coherently driven at the frequencies of the two modes. We consider as the intracavity medium (a) a two-photon absorber, and (b) $N$ three-level atoms. These systems display a new class of bifurcations, and, in the case of the two-photon absorber, optical tristability is possible.

DOI
10.1103/physreva.24.627

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We consider an intracavity nonlinear medium interacting with two modes of the electromagnetic field via a two-photon process. The cavity is coherently driven at the frequencies of the two modes. We consider as the intracavity medium (a) a two-photon absorber, and (b) NNN three-level atoms. These systems display a new class of bifurcations, and, in the case of the two-photon absorber, optical tristability is possible.
We consider a new class of bifurcations that may arise in multiphoton processes inside a coherently driven optical cavity involving more than one mode of the radiation field. For a non-saturable, non-linear medium, a single bifurcation point exists where the symmetric solution bifurcates into a number of non-symmetric solutions. The nature of the bifurcation ranges from a simple pitchfork bifurcation, in the case of four-wave mixing, to more complicated phenomena in higher-order processes. A saturable non-linear medium exhibits similar behaviour for low-input intensities; however, as the input intensity is increased the medium saturates, and at a second bifurcation point the symmetric branch regains its stability. The presence of fluctuations assures the accessibility of the symmetric branch. Thus, for example, for a two-photon absorbing medium we have the possibility of optical tristability involving one symmetric solution and two non-symmetric solutions.
We consider three nonlinear optical models that display a two-photon process and exhibit a Hopf bifurcation corresponding to a phase instability. The three models describe second-harmonic generation, two-photon optical bistability, and degenerate four-wave mixing in resonant cavities. In all three cases we consider only the resonant configuration and analyze the attractors which emerge when the driving field is increased beyond the Hopf bifurcation threshold. At a finite distance from threshold, we find a hysteresis domain that involves a pair of periodic attractors. For all three models, each of the two solutions has the same ``field portrait'' i.e., the same representation in a plane whose axes are the imaginary and the real parts of the electric field. This suggests that the scenario is generic for this class of systems. On the other hand, the models for two-photon optical bistability and degenerate four-wave mixing in resonant cavities exhibit additional nongeneric attractors when the driving field is further increased.
The theory of the optical multistability by two-photon resonant interaction is investigated. The nonlinear medium inside a Fabry-Perot cavity is in two-photon resonance with the laser beam. The two-photon equation of state including dynamical Stark shift and relaxation parameters is obtained and studied under mean-field approximation. The equations for both Fabry-Perot cavity and ring cavity take basically the same form. The physics mechanisms resulting in optical multistability are also studied. It has been shown that the combined effects of Kerr switching and saturation absorption switching can lead to two-photon optical double-bistability, tristability as well as bistability. Furthermore, the criterion conditions under which double-bistability or tristability can occur are given and discussed. Finally, the phase-conjugated reflectivity by two-photon four-wave mixing is also shown to exhibit multistability as well as hysteresis.
We performed a comprehensive study of the temporal dynamics of a two-photon laser with resonant injected signal in the high-Q cavity limit (class-A laser). The stationary solutions as well as their stability are completely characterized. In particular it is shown that tristable behaviour is possible, in contrast with two-photon optical bistability. It is also shown that there are two independent Hopf bifurcations, the sub- or supercritical character of which is analytically determined. The dynamics of the system far from the bifurcation points is classified through a numerical study.