Quasidistribution function

Negative values of this parameter indicate sub-Poissonian photon statistics, namely, nonclassical character of the field. One obvious example of the nonclassical field is a field in a number state n) for which the photon number variance is zero, and we have g 2 (0) = 1 — 1 /n and q = — 1. For coherent states, g (0) = 1 and q = 0. In this context, coherent states draw a somewhat arbitrary line between the quantum states that have classical analogs and the states that do not have them. The coherent states belong to the former category, while the states for which g (0) < 1 or q < 0 belong to the latter category. This distinction is better understood when the Glauber-Sudarshan quasidistribution function P(ct) is used to describe the field. 

The coherent states (13) can be used as a basis to describe states of the field. In such a basis for a state of the field described by the density matrix p, we can introduce the quasidistribution function P(a) in the following way ... 

Generally, according to Cahill and Glauber [10], one can introduce the, v-parametrized quasidistribution function (a) defined as... 

As we have already discussed in Section II, another characteristic of the quantum field is its phase distribution. 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 5-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]. 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. In this case we have a two-mode field which requires a modification of the formulas presented in Section II into a two-mode case. The modification is rather trivial, and for the joint probability distribution for the continuous phase variables 0 and 0/, describing phases of the two modes, we get the formula [53]... 

The Glauber-Sudarshan P representation of the field state is associated with the normal order of the field operators and is not the only ("-number representation of the quantum state. Another quasidistribution that is associated with antinormal order of the operators is the g representation, or the Husimi function, defined as... 

Although many properties of the phase CS are known by now, for their better understanding it is very useful to analyze graphs of their quasidistributions. The discrete Wigner function, as defined by Wootters [55] (see also Ref. 57), takes the following form for s > 1... 

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