Random Polarization Dynamics in a Resonant Optical Medium Katherine A. Newhall,1 Ethan P. Atkins,2 Peter R. Kramer,3 Gregor Kovaˇciˇc,3,∗ and Ildar R. Gabitov4 1Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 2 Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720-3840 3Mathematical Sciences Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180 4Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721 ∗Corresponding author: [email protected] Compiled January 8, 2013 Random optical-pulse polarization switching along an active optical medium in the Λ-configuration with spatially disordered occupation numbers of its lower energy sub-level pair is described using the idealized integrable Maxwell-Bloch model. Analytical results describing the light polarization-switching statistics for the single self-induced transparency pulse are compared with statistics obtained from direct Monte-Carlo numerical simulations. c 2013 Optical Society of America OCIS codes: 190.5530, 190.7110, 250.6715 ∗ ∗ The model of light interacting with a material sample ∂tµ = [E+ ρ− + E−ρ+ ] /2, (1d) composed of three-level active atoms has made possi- ∗ ∗ ∂t = [E+ρ+ + E+ ρ+ ble the descriptions of several nontrivial optical phenom- N − ∗ ∗ +E−ρ− + E− ρ−] /2, (1e) ena, including lasing without inversion [1], slow light [2], ∗ ∗ and electric-field polarization of solitons in self-induced ∂tn± = [E±ρ± + E± ρ±] /2. (1f) transparency [3]. Its simplest version including a non- Here, E±(x, t) are the envelopes of the electric field degenerate upper and two degenerate lower working and ρ±(x, t, λ) and µ(x, t, λ) of the medium-polarization, atomic levels — the Λ configuration — is completely in- n±(x, t, λ) and (x, t, λ) the population densities of the tegrable when the pulse width is much shorter than the ground and excitedN levels, respectively, λ the frequency ∞ medium relaxation times [4]. It describes a new type of a detuning, and g(λ) 0, with −∞ g(λ) dλ = 1, the self-induced transparency pulse, which may be a solitary spectral-line shape. The≥ “+” and “ ” transitions in- wave only asymptotically, but in general switches into teract with the left- and right-circularlyR − polarized pulse one of the two purely two-level transitions between one components, while µ is due to the two-photon transition of the lower levels and the upper level. These transitions between the ground levels. The purely two-level “+” and correspond to circularly polarized light, and the direction “ ” transitions are invariant and involve only circularly- of the switching is determined by the population sizes polarized− light. A time-conserved quantity of Eqs. (1) is of the degenerate lower levels [5, 6]. Thus, for spatially + n+ + n− = 1, where unit normalization is chosen. disordered populations, random polarization switching N The approximations made in Eqs. (1) are (i) the takes place as a light soliton travels along the material pulse-width is much longer than the light oscillation pe- sample. The integrability of the Λ configuration furnishes riod (slowly-varying envelope approximation) and much a unique opportunity to study the mechanism responsi- shorter than the relaxation time-scales in the medium, ble for this random switching and its statistical proper- and (ii) unidirectional propagation. The latter holds pro- ties exactly in the framework of a sufficiently idealized vided the interaction time of counter-propagating pulses model, which otherwise would be impossible because of is much shorter than the nonlinear-response time of the strong nonlinearity. In this letter, we both discuss the an- medium. Equations (1) are dimensionless, e.g., the speed alytical results [7] on this switching and compare them of light is c = 1. with the results of numerical simulations. If the spectral width of the pump pulse priming the Resonant propagation of ultra-short, monochromatic, ground states of the medium is much broader than the elliptically polarized light pulses through a two-level, width of g(λ), i.e., the initial populations can be consid- active medium with a doubly degenerate ground level ered homogeneous within the width of g(λ), we find the (Λ-configuration) is described by the quasi-classical two components of the soliton solution [5–7] Maxwell-Bloch system [3, 4, 6, 8] iΘ± (x,t) E±(x, t)=4iβG±(x)e sech 2β(t x)+ τx − ∞ 1 d d− 1 d+ ∂tE± + ∂xE± = ρ± g(ν)dν, (1a) + ln | +|| | + ln cosh 2τA(x) + ln | | , (2) −∞ 2 2β2 2 d− Z ∗ | | ∂tρ+ 2iλρ+ = [E+( n+) E−µ ] /2, (1b) − N − − where G±(x) = [1 tanh (2τA(x) + ln d+ / d− )] /2 − − − − ± | | | | ∂tρ 2iλρ = [E ( n ) E+µ] /2, (1c) are their amplitudes and Θ±(x, t)=2γ(t x)+ σ[x − N − − p − ± 1 A(x)] arg d± their phases, which in turn depend on the − soliton parameters γ and β. Here, 0.2 0.2 ) ) x s s ( 0.1 ( 0.1 A(x)= 0 α(ξ)dξ, (3) ψ ψ p p R 0 0 is the cumulative initial population difference α(x) along 0.5 0.5 15 the medium sample up to any given position x, which 0 100 0 10 50 5 satisfies the asymptotic condition s/π −0.5 0 x s/π −0.5 0 x lim n±(x, t, λ) = [1 α(x)] /2 0. (4) t→−∞ ± ≥ Fig. 1. PDF pψ(x; s), with β = 1/3, γ = 1/3, ε = 0, d+ = d− = i, theoretical (black lines) and results from The rest of the material variables are known to vanish as 1600 simulations (gray lines; green online). Left: b = 0, t for this solution [6], so that only the two degen- a =0.75. Right: b = 0.75, a =0.5. erate→ −∞ lower levels are populated initially. Putting the ini- − tial time at is justified because, in gases, the lifetime 0.2 0.4 of the system−∞ ranges from 10−5 to 10−3 seconds, while ) ) s − s ( 8 0.1 ( 0.2 η the typical pulse-width is 10 seconds or shorter [9]. η p The real-valued coefficients σ and τ are given by p 0 0 ∞ g(ν) 0.2 0.2 15 0 100 0 10 σ + iτ = −∞ 8(γ+iβ−ν) dν, (5) 50 5 s/π−0.2 0 x s/π−0.2 0 x with β > 0. Equation (2)R shows that the maximal ampli- tude of each soliton component equals 4β and its tem- Fig. 2. PDF pη(x; s), with β = 1/3, γ = 1/3, ε = 0, poral width equals 1/(2β). The constants d± give the d+ = d− = i, theoretical (black lines) and results from soliton phase and position. Note that, since τ < 0, the 1600 simulations (gray lines; green online). Left: b = 0, amplitude G+(x) decreases and G−(x) increases with a =0.75. Right: b = 0.75, a =0.5. increasing A(x), and vice versa with decreasing A(x), − which is the polarization-switching effect of [5, 6]. The light-pulse polarization can be described in terms light used to prepare the optical medium, where λ is of the polarization ellipse, which is characterized by p the average wavelength of the pump light and ∆λ the the orientation and ellipticity angles, ψ and η, with p characteristic width of the light-source spectral line. If π/4 η π/4. These can be found from the for- − ≤ ≤ ∗ ∗ ∗ ∗ the pump was a Ti-sapphire laser, λp 800 nm and mulas tan 2ψ = i(E+E− E−E+)/(E+E− + E−E+) ∼ 2 −2 2 2 ∆λp 5 nm [10], so ℓc 0.1 mm λ0 ( 600 nm and sin 2η = ( E+ E− )/( E+ + E− ), which, ∼ ∼ ≫ ∼ | | − | | | | | | ∗ for sodium vapor) and a several-centimeters long exper- for the soliton (2) give ψ = σA(x) + arg d−d /2, − + imental device would be sufficiently long to capture the sin 2η = tanh [2τA(x) + ln ( d / d− )] [6]. Note that + desired statistical effects. these two angles are time-independent.| | | | The orientation angle ψ(x) behaves like a Brownian If the initial population difference α(x) in the medium motion with drift σb and diffusion coefficient 1 σ2a2and is random and spatially statistically homogeneous, we − 2 its probability density function (PDF), pψ(x; s), at any x can approximate it as white noise 1 ∗ is Gaussian in s with mean ψ(x) = σbx+ 2 arg(d−d+) ′ 2 ′ 2 2 2 h i − and variance σψ(x)= σ a x. Note that the value of ψ(x) α(x) = b, [α(x) b][α(x ) b] = a δ(x x ), (6) ∗ h i h − − i − is fixed at ψ = arg(d−d+)/2 when σ = 0. The PDF where denotes ensemble averaging over all possi- pψ(x; s) shows excellent agreement with numerical sim- ble realizationsh·i of α(x), and δ( ) is the Dirac Delta ulations in Fig. 1. The Lorentzian spectral-line shape · 2 2 function. This approximation is consistent provided the g(λ) = ε/π(λ + ε ) was used in determining pψ(x; s) pulse-carrier frequency λ0, the correlation length Lc of and all subsequent PDFs. Comparisons in all figures are α(x), the soliton width 1/β, and the observation loca- made with ε = 0 corresponding to the Dirac Delta func- tion x along the sample satisfy the inequalities λ tion, i.e., the limit of an infinitely sharp spectral line. 0 ≪ Lc 1/β x. The first is related to the slowly- The PDF for the ellipticity angle η at any x equals − varying≪ envelope≪ approximation (mentioned above), the p (x; s) = (1/√2πxa τ cos2s)exp [tanh 1(sin 2s) η | | {− − second to the unidirectionality assumption, and the last 2τbx ln d / d− ]2/8a2τ 2x for π/4 s π/4.