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Additional resources for Advances in Chemical Physics, Vol.119, Part 1. Modern Nonlinear Optics (Wiley 2001)
This means that we can factor modes, which is conserved by the interaction H the quantum evolution operator ^ ^ 0t ^It ÀiHt ÀiH ÀiH exp ¼ exp exp ð114Þ h " h " "h If the Fock state basis is used to describe the field state, we find, for the initial state jn; 0i ¼ jnij0i with n photons in the fundamental mode and zero photons ^ 0 splits the Hilbert space in the second harmonic mode, that the Hamiltonian H ^ 0 is a constant of motion, we have for a given into orthogonal sectors. Since H number of photons n the relation bi ¼ n ai þ 2h^ bþ ^ h^ aþ ^ ð115Þ which implies that the creation of k photons of the second-harmonic mode requires annihilation of 2k photons of the fundamental mode.
Since the latter solutions are exact up to the fourth order, they show restricted applicability of ^ a ðtÞ2 i and h½ÁP ^ b ðtÞ2 i the linearized solutions. We see that the quadratures h½ÁQ become smaller than unity, showing squeezing, while the other two quadratures grow above unity. The symbolic calculations using a computer allows for easy derivation of the approximate formulas for any operators for the two modes. Beside squeezing it is interesting to study the variance of the photon number operator for both modes in order to look for a possibility of obtaining the sub-Poissonian photon statistics in the process of second-harmonic generation.
In this way we get a kind of phase distribution that can be considered as an approximate description of the phase properties of the field. One can calculate the s-parametrized phase distributions, corresponding to the s-parametrized quasidistributions, for particular quantum states of the field . However, a better way to study quantum phase properties is to use the Hermitian phase formalism introduced by Pegg and Barnett [11–13]. We have already introduced this formalism in Section II. Now, we apply this formalism to study the evolution of the phase properties of the two modes in the SHG process.