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

6. Quantum effects in optical fibre communications systemsTable 6.1: Fitting parameters for the 11-Lorentzian model of the Raman gain spectrum h R (ω).All frequencies are in terahertz.j F j ν j δ j0 0.16 0.005 0.0051 -0.3545 0.3341 8.00782 1.2874 26.1129 46.65403 -1.4763 32.7138 33.05924 1.0422 40.4917 30.22935 -0.4520 45.4704 23.69976 0.1623 93.0111 2.13827 1.3446 99.1746 26.78838 -0.8401 100.274 13.89849 -0.5613 114.6250 33.937310 0.0906 151.4672 8.3649term Γ G , whose correlations are defined by〈ΓG (ν, ζ)Γ ∗ G (ν′ ,ζ ′ ) 〉 = (α G + α A )δ(ζ − ζ ′ )δ(ν + ν ′ ), (6.15)2nwhere Γ G (ν, ζ) is the Fourier transform of the noise source:Γ G (ν, ζ) = 1 √2π∫ ∞−∞dτΓ G (τ,ζ)e iντ . (6.16)Similarly, the real Raman noise, which appears as a multiplicative stochastic variable Γ R ,has correlations〈ΓR (ν, ζ)Γ R (ν ′ ,ζ ′ ) 〉 = 1 n δ(ζ − ζ′ )δ(ν + ν ′ )[n th (ν)+ 1 ]α R (ν), (6.17)2where the dimensionless gain function is α R (ν) =α R (ωτ 0 )= √ 8π|I{h R (ν)}|, andthethermal Bose distribution is given by n th (ν) =[exp(ν/kTτ 0 ) − 1] −1 . Thus the Ramannoise is strongly temperature dependent, but it also contains a spontaneous componentwhich provides vacuum fluctuations even at T =0.Equations (6.11-6.17) can be discretised and, without any further approximation, canbe numerically simulated using a split-step Fourier integration routine. These equationsinclude all the currently known noise sources that are significant in soliton propagation,128