Quasi-probability Distribution - Time Evolution and Operator Correspondences

Time Evolution and Operator Correspondences

Since each of the above transformations from through to the distribution function is linear, the equation of motion for each distribution can be obtained by performing the same transformations to . Furthermore, as any master equation which can be expressed in Lindblad form is completely described by the action of combinations of annihilation and creation operators on the density operator, it is useful to consider the effect such operations have on each of the quasiprobability functions.

For instance, consider the annihilation operator acting on . For the characteristic function of the P distribution we have

Taking the Fourier transform with respect to to find the action corresponding action on the Glauber P function, we find

By following this procedure for each of the above distributions, the following operator correspondences can be identified:

Here κ = 0, 1/2 or 1 for P, Wigner and Q distributions, respectively. In this way, master equations can be expressed as an equations of motion of quasiprobability functions.