• jaynes-cummings
• liouville
• von neumann

Despite its great successes in describing the statistical aspects of experiments, quantum theory itself presents problems of interpretation, which arise from its indeterministic
features and which are clearly seen, for example, in the unresolved measurement problem.
Thus, concerning the foundations of quantum mechanics, there is an increasing impetus to
try to reconstruct and to better understand the emergence of quantum mechanics from simpler
structures beneath.

We present a reformulation of Hamiltonian dynamics in such a way that the classical Liouville
equation, after suitable tranformations, can be written in a way that
almost reproduces the quantum mechanical von Neumann equation. Indeed, an unusual
superoperator appearing in the transformed Liouville equation gives rise to the
only substantial difference between the quantum and classical dynamics. This approach also
gives the possibility of looking more carefully into the common
and distinctive features of the classical and quantum dynamics. Some interesting considerations
regarding the preparation of entangled states are pointed out.

We then present this approach for a hydrogenic atom interacting with the
electromagnetic field in the Coulomb gauge. Starting from the classical Liouville equation
we derive for the atom-field system an evolution equation appearing as the usual quantum
mechanical von Neumann equation modified by an extra term. From this general expression,
by making approximations which are widely applied in quantum optics,
we obtain the von Neumann equation for the Jaynes-Cummings model
modified by an extra term.

The Jaynes-Cummings model describes a simplified version of atom-field interaction. It has been
widely studied experimentally and theoretically. In particular, it has been a usefull tool for
investigating the evolution of entanglement between the atom and the electromagnetic field
and has also been used for applications in quantum information. A review of the related
cavity QED experiments performed in Paris is presented.

The aim of the thesis is to study the way the extra term modifies the dynamics
described by the Jaynes-Cummings hamiltonian and thus to what extent the dynamics of this
quantum mechanical model can be derived from the classical Liouville equation.