For long wavelength, compared to the size of the atom The term containing A 2 in the dipole approximation does not involve atomic operators, consequently matrix elements are zero. Known as the dipole approximation

Frequency of the atomic transition With respective energies denote the lower (ground) and upper (excited) states

≠0 is the frequency of the atomic transition For the circularly polarized radiation For linearly polarized radiation And the matrix element

Making the transformations For the slowly varying amplitudes

:We adopt the rotating wave approximation (RWA) which amounts to neglecting oscillating sum frequencies as opposed to those oscillating with Where we have used the following properties of the Laplace transforms:

Which satisfy the initial conditions, as well as the normalization condition The inverse Laplace transform of

Precession of Bloch vector R about the effective field for (a) (b)

Let us introduce the atomic operators

Defining the atom-cavity field coupling constant we can write In the interaction picture Terms, which do not conserve the total energy of the system, are dropped in the rotating wave approximation, The total Hamiltonian becomes Known as the Jaynes-Cummings model

The only non-vanishing matrix elements

If at time t = 0 the atom is in the upper state and the field state is For the atomic population inversion For exact resonance

At t = 0 the atom is in the ground state but the cavity field is in a coherent state

Time-dependence of the inversion of a two-level atom interacting with a quantum single- mode coherent field (a), and chaotic (thermal) field (b)

No revivals, completely chaotic behavior, due to the absence of any phase relations between various the inversion undergoes collapses and revivals, the time-scale depends on Each component Tends to drive the atom with its Rabi frequency