By Myron W. Evans, Ilya Prigogine, Stuart A. Rice
The recent variation will give you the sole finished source on hand for non-linear optics, together with unique descriptions of the advances during the last decade from world-renowned specialists.
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Extra info for Advances in Chemical Physics, Vol.119, Part 1. Modern Nonlinear Optics (Wiley 2001)
Classical solutions discussed earlier, ua ðtÞ ¼ sech t and ub ðtÞ ¼ tanh t, indicated that the amplitudes of the two modes are monotonic functions of time and that eventually all the energy from the fundamental mode will be transferred into the secondharmonic mode, assuming that there was no second-harmonic signal initially. It is well known [20,48], however, that the quantum solution has oscillatory character and does not allow for the complete power transfer. Using the state (127) we find that the mean photon numbers evolve in time according to the formulas ^ a ðtÞi ¼ hcðtÞj^ aþ ^ ajcðtÞi ¼ hN ^ b ðtÞi ¼ hcðtÞj^ hN bþ ^ bjcðtÞi ¼ 1 X b2n ½n=2 X ðn À 2kÞjcn;k ðtÞj2 n¼0 k¼0 1 X ½n=2 X n¼0 b2n kjcn;k ðtÞj2 ð135Þ k¼0 Because of the Poissonian factors, which are peaked at Na , the summation over n can be performed numerically if Na is not too great.
Nikitin and Masalov  suggested that the two peaks appearing in the Q function indicated a macroscopic superposition of quantum states. In case of the second harmonic mode the Q function starts with the peak localized at the origin (initial vacuum state) and moves along the Im b axis undergoing deformation during the evolution. Motion of the centroid of the distribution along the Im b axis confirms again our earlier prediction that the phases of the two fields exhibit a shift by p=2. The Q function is one of the quasiprobalility distributions that describe quantum statistical properties of the field.
It was shown by Ou  that the two systems can be solved analytically, giving pﬃﬃﬃ ^ a ð0Þð1 À t tanh tÞ sech t À ÁP ^ a ðtÞ ¼ ÁQ ^ b ð0Þ 2 tanh t sech t ÁQ ^ a ð0Þ p1ﬃﬃﬃ ðtanh t þ t sech2 tÞ þ ÁP ^ b ð0Þ sech2 t ^ b ðtÞ ¼ ÁQ ÁP 2 ^ b ð0Þ p1ﬃﬃﬃ ðsinh t þ t sech tÞ ^ a ð0Þ sech t þ ÁQ ^ a ðtÞ ¼ ÁP ÁP 2 pﬃﬃﬃ ^ b ð0Þð1 À t tanh tÞ ^ b ðtÞ ¼ ÀÁP ^ a ð0Þ 2 tanh t þ ÁQ ÁQ ð86Þ 24 ryszard tanas´ Now, assuming that the two modes are not correlated at time t ¼ 0, it is straightforward to calculate the variances of the quadrature field operators and check, according to the definition (12), whether the field is in a squeezed state.