Maxwell-Bloch Equations as Predator-Prey System

Abstract

Abstract. Regions of the full parameter space for which chaotic behavior in laser models based on the Maxwell-Bloch equation occurs are studied [1]. The range in the parameter space have been charted, all possibilities for the value of the maximal Lyapunov exponent are shown to exist, positive maximal Lyapunov exponents characterizing chaotic behavior occurs for parameter values that correspond to the range of parameters for Helium-Neon lasers. The Maxwell-Bloch equations as given by [2] and [3] involve the coupling of the fundamental cavity mode, E with the collective variables P and ∆, that represent the atomic polarization and the population inversion. They are represented by the following equations.For the parameter values k = σ, γ ⊥ = g 2 /k = 1, g 2 ∆o/k = r, γ = b, the system can be transformed into the Lorenz system about the equilibrium point ∆ = ∆ o by setting x = E, y = gP/k,z = ∆ o − ∆. The meaning of the parameters in the original equations as given by Arrechi[2], while σ, r, b are the Lorenz parameters.Maxwell-Bloch equations can be transformed into a system proposed by [4], that resembles the Lotka-Volterra problem with I representing the intensity of the laser field and N denoting population density, the dimensionless rate equations for I and N are obtained by assuming I = E 2 and the subsidary condition gP = kE∆. Starting with the first Maxwell-Bloch Equation multiplying both sides by E gives EĖ = −kE 2 + gP E,Ṗ = −γ ⊥ P + gE∆ = −γ ⊥ P + g 2 P/k The subsidary condition enables one to solve for the polarization as P = Po exp (−(γ ⊥ P − g 2 /k)t) and the third equation becomes∆ = −γ (∆ − ∆o) − 4kE