Quasi-probability Distribution - Characteristic Functions

Characteristic Functions

Analogous to probability theory, quantum quasiprobability distributions can be written in terms of characteristic functions, from which all operator expectation values can be derived. The characteristic functions for the Wigner, Glauber P and Q distributions of an N mode system are as follows:

Here and are vectors containing the annihilation and creation operators for each mode of the system. These characteristic functions can be used to directly evaluate expectation values of operator moments. The ordering of the annihilation and creation operators in these moments is specific to the particular characteristic function. For instance, normally ordered (annihilation operators preceding creation operators) moments can be evaluated in the following way from :

In the same way, expectation values of anti-normally ordered and symmetrically ordered combinations of annihilation and creation operators can be evaluated from the characteristic functions for the Q and Wigner distributions, respectively. The quasiprobability functions themselves are defined as Fourier transforms of the above characteristic functions. That is,

Here and may be identified as coherent state amplitudes in the case of the Glauber P and Q distributions, but simply c-numbers for the Wigner function. Since differentiation in normal space becomes multiplication in fourier space, moments can be calculated from these functions in the following way:

Here denotes symmetric ordering. These representations are all interrelated through convolution by Gaussian functions:

or using the property that convolution is associative