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Communications in Theoretical Physics

Dynamic Equations and Nonlinear Dynamics of Cascade Two-Photon Laser

Xie Xia, Huang Hong-Bin, Qian Feng, Yajun Zhang, Peng Yang, Qi Guan-Xiao

Published Jun 1, 2006• 2 citations
Publisher

Abstract

We derive equations and study nonlinear dynamics of cascade two-photon laser, in which the electromagnetic field in the cavity is driven by coherently prepared three-level atoms and classical field injected into the cavity. The dynamic equations of such a system are derived by using the technique of quantum Langevin operators, and then are studied numerically under different driving conditions. The results show that under certain conditions the cascade two-photon laser can generate chaotic, period doubling, periodic, stable and bistable states. Chaos can be inhibited by atomic populations, atomic coherences, and injected classical field. In addition, no chaos occurs in optical bistability.

DOI
10.1088/0253-6102/45/6/018

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We derive equations and study nonlinear dynamics of cascade two-photon laser, in which the electromagnetic field in the cavity is driven by coherently prepared three-level atoms and classical field injected into the cavity. The dynamic equations of such a system are derived by using the technique of quantum Langevin operators, and then are studied numerically under different driving conditions. The results show that under certain conditions the cascade two-photon laser can generate chaotic, period doubling, periodic, stable and bistable states. Chaos can be inhibited by atomic populations, atomic coherences, and injected classical field. In addition, no chaos occurs in optical bistability.
Based on the cascade two-photon laser dynamic equation derived with the technique of quantum Langevin operators with the considerations of coherently prepared three-level atoms and the classical field injected into the cavity, we numerically study the effects of atomic coherence and classical field on the chaotic dynamics of a two-photon laser. Lyapunov exponent and bifurcation diagram calculations show that the Lorenz chaos and hyperchaos can be induced or inhibited by the atomic coherence and the classical field via crisis or Hopf bifurcations.
We derive the dimensionless dynamic equations of two-photon lasers with Λ atomic level configuration by using the quantum Langevin equation method with the considerations of atomic coherence and injected classical fields. Then we analyze the stability and the chaotic dynamics of the two-photon laser by calculating the bifurcation diagram and the maximum Lyapunov exponent (MLE). Our results show that the Lorenz strange attractors and one-focus strange attractors can exist in this system, and the chaos can be induced or inhibited by the injected classical fields via Hopf-bifurcations or crises, while the atomic coherence induces chaos via crises, and inhibit chaos via Hopf bifurcation or crises.
The dynamics of an ensemble of two-level atoms injected into a single-mode cavity is studied in the exact atom-field interaction situation, in which the counter-rotating terms describing the so-called virtual photon processes neglected in the rotating-wave approximation, are considered. The cavity mode is driven by the injected classical field,and the atom is prepared in a coherent superposition of the two levels. We first derive the generalized Lorenz-Haken equation by using the technique of quantum Langevin equation, and then numerically study the dynamics of this equation.We find that the virtual photon processes have strong effects on the dynamics, which can cause the trajectory in phase space of strange attractor spiral around four focus points, and the trajectory is modulated by virtual photon processes.The chaos region in parameter space is now enlarged. It should be stressed that the strange attractor can exist in optical bistability, and whether the atomic coherences and classical field can inhibit chaos depends on the laser frequency.
The dynamics of an ensemble of two-level atoms injected into a single-mode cavity is studied in the exact atom-field interaction situation, in which the counter-rotating terms describing the so-called virtual photon processes neglected in the rotating-wave approximation, are considered. The cavity mode is driven by the injected classical field, and the atom is prepared in a coherent superposition of the two levels. We first derive the generalized Lorenz–Haken equation by using the technique of quantum Langevin equation, and then numerically study the dynamics of this equation. We find that the virtual photon processes have strong effects on the dynamics, which can cause the trajectory in phase space of strange attractor spiral around four focus points, and the trajectory is modulated by virtual photon processes. The chaos region in parameter space is now enlarged. It should be stressed that the strange attractor can exist in optical bistability, and whether the atomic coherences and classical field can inhibit chaos depends on the laser frequency.
We present nonlinear dynamical features of two-photon double-cavity optical bistability exhibited by a three-level ladder system in the mean-field limit at low input light levels. The system exhibits a hump-like feature in the lower branch of the bistable response, wherein a region of instability develops. The system displays a range of dynamical features varying from normal stable switching to periodic self-pulsing and chaos. The inclusion of two competing cooperative atom-field couplings leads to such rich, nonlinear dynamical behavior. We provide a domain map that clearly delineates the various regions of stability, as well as bifurcation diagrams with associated supporting evidence that identifies the period-doubling route to chaos.
The cooperative behavior of a system of atoms contained in a ring cavity is investigated under the condition that the fundamental atomic transition is a two photon transition1. Maxwell-Block equations are used to obtain the input vs. output relation by adiabatically eliminating all the atomic variables. The fundamental equation that results is a complex two dimensional map--which in the mean field limit reduces to the standard bistability equations. The characteristics of the two dimensional map are numerically investigated. Such a map is shown to lead to chaotic behavior following the Feigenbaum2 scenario. The sequence of events following first regime of chaos is a set of period halving bifurcations. The characteristics of the power spectrum in the region of chaos are presented. The effect of noise on the period doubling bifurcations in the present model, is also investigated and the connection with the theoretical predictions3 is established. The changes in the dynamical characteristics of the system when atomic inversion relaxes on the same time scale as the cavity round trip time will also be discussed in detail.