The phase distribution of the quantum field can be calculated from the quasidistribution functions by integrating over the radial variable. 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 [16]. However, a better way to study quantum phase properties is to use the Hermitian phase formalism introduced by Pegg and Barnett [11–13].

5a we have plotted the normalized second-harmonic intensity ^ b i=Na , where Na is the initial mean number of photons of the coherent nb ¼ 2hN state of the fundamental mode, against the scaled time t for the initially coherent state with the mean number of photons equal to 2 (solid line) and compared it with the corresponding intensity obtained for the initial state of the fundamental mode being the Fock state with two photons [Eq. (132)]. In the latter case we see the perfectly periodic behavior with complete transfer of energy between the fundamental and second-harmonic modes.

1 both uncertainty products are divergent as t2 =2. The evolution of the uncertainty products is illustrated in Fig. 3b. Since, except for the initial value, the value of the uncertainty product is larger than unity, the quantum states produced in the second-harmonic generation process are not the minimum uncertainty states. The linear approximation to the quantum noise equations presented in this section shows that even in linear approximation the inherent property of quantum fields — the vacuum fluctuations which are ubiquitous and always present — undergo essential changes when transformed nonlinearly.