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.

Read more about this topic:  Quasi-probability Distribution

Famous quotes containing the words time and/or evolution:

    “In time, after a dozen years of centering their lives around the games boys play with one another, the boys’ bodies change and that changes everything else. But the memories are not erased of that safest time in the lives of men, when their prime concern was playing games with guys who just wanted to be their friendly competitors. Life never again gets so simple.”
    —Frank Pittman (20th century)

    “By contrast with history, evolution is an unconscious process. Another, and perhaps a better way of putting it would be to say that evolution is a natural process, history a human one.... Insofar as we treat man as a part of nature—for instance in a biological survey of evolution—we are precisely not treating him as a historical being. As a historically developing being, he is set over against nature, both as a knower and as a doer.”
    —Owen Barfield (b. 1898)