Stochastic pulse switching in a degenerate resonant optical medium

Abstract

Using the idealized integrable Maxwell-Bloch model, we describe random optical-pulse polarization switching along an active optical medium in the Λ-configuration with disordered occupation numbers of its lower energy sub-level pair. The description combines complete integrability and stochastic dynamics. For the single-soliton pulse, we derive the statistics of the electric-field polarization ellipse at a given point along the medium in closed form. If the average initial population difference of the two lower sub-levels vanishes, we show that the pulse polarization will switch intermittently between the two circular polarizations as it travels along the medium. If this difference does not vanish, the pulse will eventually forever remain in the circular polarization determined by which sub-level is more occupied on average. We also derive the exact expressions for the statistics of the polarization-switching dynamics, such as the probability distribution of the distance between two consecutive switches and the percentage of the distance along the medium the pulse spends in the elliptical polarization of a given orientation in the case of vanishing average initial population difference. We find that the latter distribution is given in terms of the well-known arcsine law.

Random polarization dynamics in a resonant optical medium

“…The integrability of the Λ configuration furnishes a unique opportunity to study the mechanism responsible for this random switching and its statistical properties exactly in the framework of a sufficiently idealized model, which otherwise would be impossible because of strong nonlinearity. In this letter, we both discuss the analytical results [17] on this switching and compare them with the results of numerical simulations.…”

“…More precisely, we study the statistics of the following random distances: X tra , over which the pulse polarization evolves from linear (η = 0) to nearly-circular of either orientation (|η| = η κ ); X int , over which the pulse polarization evolves from linear to nearly-circular of either orientation and back to linear; and X dep , beyond which the pulse polarization remains forever circularly-polarized, |η| > η κ , for all greater distances. When b = α(x) = 0 we find [17] p…”

“…If the spectral width of the pump pulse priming the ground states of the medium is much broader than the width of g(λ), i.e., the initial populations can be considered homogeneous within the width of g(λ), we find the two components of the soliton solution [15,16,17]…”

Polarization dynamics in a resonant optical medium with initial coherence between degenerate states

“…The electric field solutions are used to demonstrate the polarization switching phenomenon explained in the introduction. The polarization switching phenomenon has been investigated in [8] in the absence of initial coherence between the sublevels (µ 0 = 0) and in [3,22] for the case of random initial populations in the lower two energy levels. These spatially disordered populations are modeled by allowing the population difference α = n + − n − in the initial population matrix ρ 0 in (4) to be a random function of the distance into the material, x.…”

“…and we see that P solves the same equation that Ψ 0 solves with λ = λ 0 , and therefore must be a constant multiplier of this known solution. The known solution Ψ 0 is trivial to find, as it solves the constant-coefficient equations in (3). It is given by…”