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

6. Quantum effects in optical fibre communications systemsnumber dependence of the angular frequency ω:v = ∂ω∣ ∂k ∣ , ω ′′ = ∂2 ω ∣∣∣k=kck=kc∂k 2 . (6.4)The nonlinear term in Eq. (6.3) arises from the intensity dependent refractive indexn = n 0 + In 2 . It is often called the χ (3) effect, so named because it arises from the thirdorderterm in the expansion of the polarisation field in terms of the electric field[1]. Thecoefficient of the electronic nonlinearity is defined asχ e = (1 − f)n 2ωc 2v2, (6.5)Acwhere ω c is the carrier frequency, A is the effective cross-sectional area of the travelingmode, and f is the fraction of the nonlinearity due to the Raman gain:∫ ∞f = χ R= 2 ∫ ∞dt dωr 2 (ω)sin(ωt)χ e χ e 0 0≃ 0.2, (6.6)where the numerical value for f has been determined from a detailed fit to the experimentallymeasured Raman gain spectrum, details of which will be given later.In the last term of Eq. (6.3), the atomic vibrations within the silica structure of thefibre are modelled as a continuum of localised oscillators, and are coupled to the radiationmodes by a Raman transition with a frequency-dependent strength of r(ω). Theatomic displacement is proportional to Â+† , where the phonon annihilation and creationoperators,  and † , have the commutation relations[Â(x, ω),  † (x ′ ,ω )]′ = δ(x − x ′ )δ(ω − ω ′ ). (6.7)The frequency dependence of the coupling r(ω) has been determined empirically throughmeasurements of the Raman gain spectrum[23].Even though it has reduced dimensionality and lacks any spatially modulated terms,the fibre Hamiltonian Eq. (6.3) contains some complications not present in the atom opticsHamiltonian Eq. (2.1): it contains terms describing a finite group velocity and the Ramaninteraction. However, in a propagating frame of reference that moves with the radiationfield, the group velocity terms disappear in the Heisenberg equations of motion. Also,once we have integrated over the phonon reservoirs, the Raman terms contribute only anextra time-delayed nonlinearity and thermal noise sources to the equations of motion.125