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Acta Physica Sinica

Dynamics of nonphase-locking Lorenz-Haken equation

R. Ju, Zhang Ya-Jun, Huang Hong-Bin, Zhao Huan, 东南大学物理系，南京 210096

Published Jan 1, 2004• 2 citations
Publisher

Abstract

Considering the atomic coherences and injected classical field, we derived the n-Onphase-locking Lorenz-Haken equation by using the technique of stochastic differential equations. The effects of detuning, injected field and the atomic coherence on the dynamical characteristics of this equation is then numerically studied. The results show that in laser working situations, the detuning can result in chaotic behaviors of the light phase. Under different conditions, the system can generate four attractors, double attractors and single attractor, and the fractal dimension of the system is larger than that of the locking-phase system. The optical detuning, injected field and atomic coherence can inhibit the chaos of the cavity field. In the optical bistability regions, we obtain symmetric bistability pair, but never find the chaotic attractors.

DOI
10.7498/aps.53.2191

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Considering the atomic coherences and injected classical field, we derived the n-Onphase-locking Lorenz-Haken equation by using the technique of stochastic differential equations. The effects of detuning, injected field and the atomic coherence on the dynamical characteristics of this equation is then numerically studied. The results show that in laser working situations, the detuning can result in chaotic behaviors of the light phase. Under different conditions, the system can generate four attractors, double attractors and single attractor, and the fractal dimension of the system is larger than that of the locking-phase system. The optical detuning, injected field and atomic coherence can inhibit the chaos of the cavity field. In the optical bistability regions, we obtain symmetric bistability pair, but never find the chaotic attractors.
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.
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.
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.
The influence of injected atomic coherence on a laser's operation is discussed by analyzing the semiclassical equations of motion of this system. Both stationary-steady-state and time-dependent regimes of operation are investigated. We present the dependence of the stationary laser intensity and phase on the external parameters (population inversion, amplitude of atomic coherence, and detuning). For small cavity-field detuning and large atomic coherence, both frequency and phase locking occurs. The laser frequency is locked to the atomic frequency, and the phase is locked to a value determined primarily by the phase of the atomic coherence. Under certain conditions, in the inverted regime, the output intensity is an ssS-shaped function of the atomic coherence, leading to the possibility of a bistable behavior. However, only a portion of the bistable curve gives stable stationary operation, and in critical points (the location of which depends crucially on the external parameters) time-dependent instabilities branch away. When the detuning is larger than a critical value there is neither phase nor frequency locking. We find that the time-dependent behavior of the laser intensity in this case is oscillatory (quasiperiodic), and show that there is a stable limit cycle in the phase plane of the quadratures. There is a small parameter region where a stationary steady state and an oscillatory state may coexist. We also consider the nonlinear quantum theory of a laser with injected atomic coherence, and include the effect of pumping statistics. We derive the Fokker-Planck equation for the P representation, and express the noise in terms of moments. We find that in the steady state the intensity noise can be suppressed below the shot-noise limit but that the phase fluctuations are not affected by pump regularity. For nonzero detuning, we find that transient squeezing of the phase fluctuations is possible.