By Cox R.T., Hubbard J.C.

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In general, quantum noise effects are an inseparable part of nonlinearity and gain in waveguides and fibers, and give an ultimate limit to channel capacity. In summary, the results given here hold for single-mode optical fibers and waveguides, with the inclusion of: nonlinearity, birefringence, Raman and Brillouin scattering, dispersion, linear gain, and (small) losses, to give a complete quantum description. 62 CHAPTER 7. PHASE-SPACE DISTRIBUTIONS Chapter 8 Quantum solitons An optical soliton in a fiber can be described classically using the nonlinear Schr¨odinger equation[13, 14] , which is applicable to single mode propagation in nonlinear dispersive fibers or waveguides.

This needs to be extended to include the effects of thermal and quantum fluctuations due to Raman scattering. Although previous quantum treatments of Raman scattering have been given[51],[52] , it is necessary to modify these somewhat in the present situation. The Raman interaction energy[53, 54] of a fiber, in terms of atomic displacements from their mean lattice positions, is known to have the form: 1 2 WR = j . ηjR .. 1) xj ) is the electric displacement at the j-th mean atomic location Here D(¯ ¯ j , δxj is the atomic displacement operator, and ηjR is a Raman coupling x tensor.

This can be Fourier transformed, giving γs (ω) = γs (ω) + iγs (ω), where γ G > 0 for gain, and γ A > 0 for absorption. Similarly, χss uv is a ˆG is the linear quantum noise tensor response function for the nonlinearity, K 52 CHAPTER 6. RESERVOIRS ˆRss is the Raman noise, including scattering from due to gain and loss, while K one polarization into the orthogonal one. The equation can be easily generalized to include nonlinear absorption or laser saturation effects, relevant to amplifiers with intense fields, but these terms are omitted here for simplicity.