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Additional resources for Advances in Chemical Physics, Vol.119, Part 1. Modern Nonlinear Optics (Wiley 2001)
8 8 4 6 2 þ2 ^2 ^ hb b iðtÞ ¼ ja0 j ðktÞ À ðktÞ ðja0 j þ 1Þ þ Á Á Á ð102Þ 3 quantum noise in nonlinear optical phenomena 31 and combining equations (100) and (102) we obtain 4 aþ ^ a2 iðtÞ À h^ ai2 ðtÞ ¼ À2ðktÞ2 ja0 j4 þ ðktÞ4 ja0 j4 ð6ja0 j2 þ 1Þ þ Á Á Á h^ aþ2 ^ 3 4 6 8 þ2 ^2 þ^ 2 ^ ^ hb b iðtÞ À hb bi ðtÞ ¼ À ðktÞ ja0 j þ Á Á Á 3 ð103Þ The results (103), obtained first by Kozierowski and Tanas´ , explain a very important property of the second harmonic generation, that is, the appearance of the sub-Poissonian photon statistics, which is an effect of quantum properties of the fields.
Depending on the value of 4 Á ¼ E2 E 2 À 27 ð71Þ the polynomial has three different real roots (Á < 0) and two real roots, one of which is double (Á ¼ 0). The third case with Á > 0, in which the polynomial has one real root and two complex conjugate roots, is excluded on physical 4 . grounds since E2 27 4 Þ, we In case of three different real roots nb1 < nb2 < nb3 ðÁ < 0 or E2 < 27 can effect a substitution nb ¼ nb1 þ ðnb2 À nb1 Þ sin 2 f ð72Þ quantum noise in nonlinear optical phenomena 19 which leads to the elliptical integral ð ð dnb 2 df qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ nb3 À nb1 1 À k2 sin 2 f nb ð1 À nb Þ2 À E2 ð73Þ where k2 ¼ nb2 À nb1 nb3 À nb1 ð74Þ and we get from (68) and (73) ð pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ df nb3 À nb1 t ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 À k2 sin 2 f ð75Þ Using the definitions of the Jacobi elliptic functions we have sin f ¼ sn Àpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Á nb3 À nb1 t j k2 ð76Þ and inserting (76) into (72) we obtain the solution nb ðtÞ ¼ nb1 þ ðnb2 À nb1 Þ sn2 Àpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Á nb3 À nb1 t j k2 ð77Þ where sn is the Jacobi elliptic function sinusamplitude.
When the eigenvalues are numbered from the lowest to the highest 36 ryszard tanas´ value, there is an additional relation Ukj U0j ¼ ðÀ1Þk Uk;½n=2Àj U0;½n=2Àj ð123Þ which makes the coefficients cn;k ðtÞ either real (k even) or imaginary (k odd). This property of the coefficients cn;k ðtÞ is very important and allows in some cases to get exact analytical results. Knowing the coefficients cn;k ðtÞ the resulting state of the field (119) in the particular sector can be written, for the initial state jn; 0i, as jcðnÞ ðtÞi ¼ ½n=2 X ðnÞ cn;k ðtÞjck i ð124Þ k¼0 The typical initial conditions for the second-harmonic generation are a coherent state of the fundamental mode and the vacuum of the second-harmonic mode.