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

6. Quantum effects in optical fibre communications systemsfor a bright soliton solution (Eq. (6.19)). To obtain the fluctuations in velocity, we candifferentiate both sides of Eq. (6.37) and substitute in for dφ/dζ from Eq. (6.18) to getdAdζV dAdζ + AdV dζ{∫ ∞= RCombining these two equations and integrating gives}dτφ ∗ s(6.38a)=−∞{∫ ∞}−I dτφ ∗ τ s .−∞(6.38b){∫ ζ ∫ ∞}∆V (ζ) =∆V (0)+R dζ ′ dτ(−i)Aτ tanh(Aτ)sech(Aτ)e −iVτ−iθ s(τ,ζ ′ ) , (6.39)0 −∞which is the same as Eq. (6.32) and which can be used to obtain the leading order termsin the timing jitter.6.5.2 Dark solitonsFibres in the normal dispersion regime can support dark soliton solutions, so called sincethey correspond to a dip in the background intensity[78]:φ t (τ,ζ) = E √ 1 − A 2 sech 2 (EAτ − q(ζ))e iθ(ζ) e iσ(ζ,τ) , (6.40a)()A tanh(EAτ − q(ζ))σ(ζ,τ) = arcsin √ , (6.40b)1 − A 2 sech 2 (EAτ − q(ζ))where dθ/dζ = E 2 , dq/dζ = A √ 1 − A 2 E 2 and E is the amplitude of the background field.The size of the intensity dip at the centre of the soliton is given by A, with the intensitygoing to zero in a black tanh(τ) soliton, for which A = 1. Dark solitons are classed astopological solitons, because they connect two background pulses of different phase. Thetotal phase difference between the boundaries is ψ =2arcsin(A).The nonvanishing boundary conditions of the dark pulse complicate the perturbationcalculation of jitter variance. In order to use the conservation laws to calculate the growthin fluctuations, the constants of motion must be regularised so that their variational derivativesvanish at the boundary[102, 171]. The regularised momentum is I reg = I − ψ, whichenables the leading order terms in the jitter growth to be calculated in the same manneras given above. We may rewrite the dark soliton solution as( √ )φ t (τ,ζ) = Ee iθ(ζ) 1 − A 2 + Ai tanh(EAτ − q(ζ)) , (6.41)133