By Myron W. Evans, Ilya Prigogine, Stuart A. Rice
The recent version will give you the sole finished source to be had for non-linear optics, together with distinctive descriptions of the advances during the last decade from world-renowned specialists.
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Additional info for Advances in Chemical Physics, Vol.119, Part 1. Modern Nonlinear Optics (Wiley 2001)
However, for the initial conditions representing the purely second-harmonic generation process, specifically, under the assumption that the harmonic mode is initially in the vacuum state such that ^ bj0i ¼ 0, we can drop all the terms containing operators ^b or ^bþ because they give zero due to the normal ordering of the operators. Assuming, moreover, that the pump beam is in a coherent state ja0 i we find the following expansions for the mean values of the operators ^aðtÞ and ^bðtÞ  ! 1 2 2 4 4 2 h^ aðtÞi ¼ a0 1 À ðktÞ ja0 j þ ðktÞ ð5ja0 j þ ja0 j Þ þ Á Á Á 6 !
This shows that quantum fluctuations of various physical quantities are related to each other, but this relation is not necessarily simple. The statistical properties of the quantum fields are well characterized by the quasiprobability distribution functions defined in the Section II. Let us consider quantum noise in nonlinear optical phenomena 45 0 −2 Correlations −4 −6 −8 −10 −12 0 2 4 6 8 10 12 14 16 τ ^ a ÁN ^ b i=Na between the photon-number fluctuations in the two Figure 10. Correlations 4hÁN modes for Na ¼ 10 (solid line) and Na ¼ 40 (dashed line).
The distribution (142) becomes narrower as s increases approaching unity. The Q function is the broadest distribution, but all of them for s < 1 are just Gaussians with various variances. In terms of classical description of the field noise, the most suitable function is the Wigner function , for which the variance of the Gaussian distribution is equal to 1 1 4. The Q function is broader, with the variance equal to 2, but because the state of the field has a large coherent component, the two functions are very similar in shape and the Q function is usually easier to calculate.