Stochastic pulse switching in a degenerate resonant optical medium

Using the idealized integrable Maxwell-Bloch model, we describe random optical-pulse polarization switching along an active optical medium in the $\Lambda$ configuration with disordered occupation numbers of its lower-energy sublevel 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 sublevels 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 sublevel 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 arc sine law.

Figure 1

Polarization ellipse in the plane perpendicular to the direction of light propagation.

Figure 2

The two circular-polarization amplitudes, $E_{+}$ and $E_{-}$, of a real-valued, polarized-light soliton pulse propagating through a randomly prepared $\Lambda$-configuration optical medium, as described by Eq. (15a). The model of the initial population difference $\alpha \left(x\right)$ was sampled from the properly rescaled $\beta$ distribution with mean $b=\langle \alpha \left(x\right)\rangle =0.001$, variance ${\sigma }_{\alpha }^2=0.71$, and coherence length $L_{\mathrm{c}}=1.8$. The width of the Lorentzian spectral line in Eq. (19) is $\varepsilon =0.1$. The soliton parameters are $\beta =1/3$, $\gamma =0$, $d_{+}=d_{-}=i$.

Figure 3

Probability density function $p_{\mathcal{T}}(t;x)$, with $\beta =1$, $\gamma =1$, $\varepsilon =0.1$, $d_{+}=i$, $d_{-}=i/3$, $a=3$, and $b=0.8$.

Figure 4

Probability density function $p_{\psi }(s;x)$, with $\beta =1$, $\gamma =1$, $\varepsilon =0.1$, $d_{+}=i$, $d_{-}=i/3$, and $a=1$. Top: no bias, $b=0$. Bottom: nonzero bias, $b=0.3$.

Figure 5

Probability density function $p_{\eta }(s;x)$, with $\beta =1$, $\gamma =1$, $\varepsilon =0.1$, $d_{+}=i$, $d_{-}=3i/4$, and $a=1$. Top: no bias, $b=0$. Bottom: nonzero bias, $b=0.3$.

Figure 6

Distributions $p_{X_{\mathrm{tra}}}\left(x\right)$ (solid line) and $p_{X_{\mathrm{int}}}\left(x\right)$ (dashed line) for a medium with no bias, $b=\langle \alpha \left(x\right)\rangle =0$, computed using Eqs. (57) with $L_{\mathrm{tra}}=4.53$. Note the faster decay of the distribution $p_{X_{\mathrm{tra}}}\left(x\right)$. Inset: Distributions $p_{X_{\mathrm{tra}}}\left(x\right)$ (solid line) and $p_{X_{\mathrm{fluc}}}\left(x\right)$ (dashed line) for a medium with strong bias, computed using Eqs. (63) and (66), respectively, with $a=1$, $b=-0.5$, and $L_{\mathrm{tra}}=4.26$.

Figure 7

Polarization state in a medium without bias under the Wiener-process approximation ($a=1.00$, $b=0$). The length scales for the transitions and distances between switches are here $L_{\mathrm{tra}}=L_{\mathrm{int}}=4.53$, but note that the distance between switches has higher probability to take values large compared to this typical length scale.

Figure 8

Realizations of the soliton polarization for a medium with strong bias ($a=0.1$, $b=-0.5$, $L_{\mathrm{lin},0}=0.64$, $L_{\mathrm{tra}}=4.26$, $L_{\mathrm{fluc}}=0.04$) on the top and a medium with not-so-strong bias ($a=0.3$, $b=-0.5$, $L_{\mathrm{lin},0}=0.64$, $L_{\mathrm{tra}}=4.26$, $L_{\mathrm{fluc}}=0.36$) on the bottom.

Figure 9

Realizations of soliton polarization for a medium with bias becoming progressively weaker from the top to bottom panel. Top: $a=0.5$, $b=-0.5$, $L_{\mathrm{lin},0}=0.64$, $L_{\mathrm{tra}}=4.26$, $L_{\mathrm{fluc}}=0.04$. Middle: $a=1.0$, $b=-0.5$, $L_{\mathrm{lin},0}=0.64$, $L_{\mathrm{tra}}=4.26$, $L_{\mathrm{fluc}}=4.00$. Bottom: $a=3.0$, $b=-0.5$, $L_{\mathrm{lin},0}=0.64$, $L_{\mathrm{tra}}=4.26$, $L_{\mathrm{fluc}}=36.0$.