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

6. Quantum effects in optical fibre communications systemsFrom this we can calculate the growth of the fluctuations in velocity:∫ ζ∆V (ζ) = ∆V (0) + dζ ′ S V (ζ ′ )0{∫ ∞} ∫ ζ= R ∆φ(τ,ζ)f ∗ V dτ + dζ ′ S V (ζ ′ ). (6.32)−∞0Using the various noise correlations from Eqs. (6.14,6.15,6.17), the correlations in thevelocity fluctuations can be calculated:〈∆V (ζ)∆V ∗ (ζ ′ ) 〉 = 〈 ∆V (0)∆V ∗ (0) 〉 ∫ ζ+ dζ ′′ dζ ′′′ 〈 S V (ζ ′′ )SV ∗ (ζ ′′′ ) 〉0 0= A ( )6n + αG A3n + 2A2 I(t 0 )ζ ζ < ζ ′ , (6.33)nwhere the overlap integral I(t 0 ) is defined asI(t 0 )=∫ ∞ ∫ ∞−∞−∞∫ ζ ′dτdτ ′ tanh(τ)sech 2 (τ) tanh(τ ′ )sech 2 (τ ′ ) ˜F(τ/A− τ/A ′ ). (6.34)Here ˜F(τ) is the inverse Fourier transform of the fluorescence F(ν) = 1 2 (n th(ν)+ 1 2 )α R(ν).The correlations in position fluctuations correspond to the jitter in arrival times, becausewe have chosen a propagative reference frame. The jitter therefore feeds off positionfluctuations as well as noise entering through the velocity:〈∆q(ζ)∆q ∗ (ζ ′ ) 〉 = 〈 ∆q(0)∆q ∗ (0) 〉 +∫ ζThus the timing jitter is0∫ ζ ′0dζ ′′ dζ ′′′ (A 2 〈 ∆V (ζ ′′ )∆V ∗ (ζ ′′′ ) 〉+ 〈 S q (ζ ′′ )S ∗ q (ζ ′′′ ) 〉) ζ