Expectation values, uncertainties, and variances are the keys to classifying different photon statistics. The expectation value of an operator, , is found by
as described in the two--state atom chapter. The variance of an operator is the square of the uncertainty in an operator, . The variance of an operator, , is found by
The calculation of each of these entities begins with the expectation value. Finding this value with eq:q13 is relatively simple. The procedure is equivalent to finding a matrix element for the combined operator and summing over the diagonal of this matrix. This operation can be expressed symbolically as
Since every operator of interest can be expressed in terms of the creation and destruction operators, this matrix element is not difficult to uncover.
We will examine three principle operators in order to understand a quantum field. The most important is , the photon number operator. Its expectation value is the average number of photons in the field at some given time.
The other two operators are and , which are directly proportional to the electric and magnetic fields. These two operators bear the statistical information about the electric and magnetic fields. However, they do not take into account the spatial dependence of the fields, which is quite important for a full understanding of how the electric and magnetic fields exist in the cavity.
The expectation value, variance, and uncertainty of each of these operators is required to realize a full view of this system. Each may be plotted in FieldApp, which also offers the capability of plotting the electric and magnetic field inside a cavity in scaled spatial units.
Another tool that is helpful in analyzing the evolution of the field is phase space. Classically the phase space for a system is provided by a plot of q versus p; however, a quantum field is inherently non--classical. q and p are related by the uncertainty principle, . For every specific p or q value there is a distribution of measurable values for these variables. Adding this distribution to the phase space plot provides an extra dimension of information about the system; indeed, without this addition the meaning of a squeezed quantum state cannot be visualized. This expanded phase space is called the Wigner function.