Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

6. Quantum effects in optical fibre communications systemsIn order to select out the evolution of particular parameters, we need to use the adjointfunctions f Pithat obey the orthogonality condition{∫ ∞}∗R dτf Pi f Pj = δ i,j . (6.26)−∞Appropriate adjoint functions aref A = φ, (6.27a)f q = τφ, (6.27b)f V = i tanh(Aτ − q)φ, (6.27c)f θ = iτ tanh(Aτ − q)φ. (6.27d)Substituting the Taylor expansion (Eq. (6.24)) into the linearised equation (Eq. (6.22))and using the adjoint functions to project out particular parameters shows that the growthof fluctuations in the fundamental soliton parameters is governed by∂∂ζ ∆q(ζ) = A∆V (ζ)+S q(ζ), (6.28a)∂∂ζ ∆V (ζ) = S V (ζ), (6.28b)where we have taken the unperturbed velocity to be zero: V = 0. The stochastic termsare defined as{∫ ∞}S Pi (ζ) =R dτf ∗ p (ζ)s(τ,ζ) . (6.29)−∞We wish to find the growth of fluctuations in position q(ζ). Because the positiondepends on the soliton frequency V , as seen from Eq. (6.19), the contributions arisingfrom both S q and S V must be considered. Firstly,{∫ ∞(S q (ζ) = R dτAτsech(Aτ − q)e −iVτ−iθ Γ G +iΓ R Asech(Aτ − q)e iVτ+iθ)}−∞∫ ∞= dτAτsech(Aτ − q)R{e −iVτ−iθ Γ G }, (6.30)and−∞{∫ ∞S V (ζ) = R=∫ ∞−∞( dτAτ(−i)sech(Aτ − q) tanh(Aτ − q)e −iVτ−iθ Γ G +iΓ R Asech(Aτ − q)e iVτ+iθ)}−∞dτAτsech(Aτ − q) tanh(Aτ − q)()Asech(Aτ − q)Γ R + I{e −iVτ−iθ Γ G } . (6.31)131