This Hamiltonian popped up when I was reading an article, as a reference(wikipedia): http://en.wikipedia.org/wiki/Jaynes–Cummings_model#cite_note-1 I don't understand why the Hamiltonian [itex]\hat H_{atom}[/itex] and [itex]\hat H_{int}[/itex] look the way they are. Usually we we just take a classical Hamiltonian and "operatorize" it, but I fail to see the classical counterpart for [itex]\hat H_{atom}[/itex] and [itex]\hat H_{int}[/itex]
[itex]\hat H_{atom}[/itex] is just a generic two-level system. Therefore in the easiest case you get just two energy levels at [itex]E_{1/2}=\pm\hbar \omega[/itex]. The interaction Hamiltonian just describes the interaction between the bosonic light field and this two-level system. The energy-conserving terms of [itex]\hat E \hat S[/itex] describe the destruction of a photon combined with the simultaneous excitation of the two-level system and the creation of a photon combined with the simultaneous transition of the two-level system from the excited to the ground state.
I can't see the physics you said form [itex]\hat E \hat S[/itex]. Actually I did find something more elaborate on this: http://uncw.edu/phy/documents/Shafer499Talk.pdf But I don't understand the content on page 27, i.e. why are those four equations the defining properties of a dipole operator, the author gave a handwaving reason "The dipole operator is responsible for “moving” the atom between energy levels.", but I don't really see why it has to be the way it is. EDIT: I think I understand now, the author is probably referring to dipole transition.