Here we give an introductory account of the wave kinetic theory. Its basic ingredients are the Wigner function and its evolution equation. Historically it started in 1932, when Wigner proposed his function as a way to represent the quantum state of a particle in its classical phase space . Later, in 1949, Moyal was ableto derive an exact evolution equation for the Wigner function, starting from the Shrödinger equation . In the classical limit, this evolution equation tends to the classical single particle Liouville equation. With these two pieces of knowledge, we are able today to build-up a consistent description of quantum particles in self consistent mean-fields, which are very useful to describe many different processes in quantum gases, namely, elementary excitations, collective processes and resonant interactions, as shown through many different examples in this book. This wave kinetic description has been abundantly used in the literature, and in particular for laser cooling, as discussed in the reviews [3, 4].
Wigner Function Laser Field Density Matrix Element Weyl Transformation Classical Phase Space
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