For long wavelength, compared to the size of the atomFor long wavelength, compared to the size of the atom

The term containing AThe term containing A22 in the dipole approximation does not involve atomic in the dipole approximation does not involve atomic operators, consequently matrix elements are zero.operators, consequently matrix elements are zero.

Known as the dipole approximationKnown as the dipole approximation

Frequency of the atomic transitionFrequency of the atomic transition

With respective energiesWith respective energies

denote the lower (ground) and upper (excited) statesdenote the lower (ground) and upper (excited) states

≠≠0 0

is the frequency of the atomic transitionis the frequency of the atomic transition

For the circularly polarized radiationFor the circularly polarized radiation

For linearly polarized radiationFor linearly polarized radiation And the matrix elementAnd the matrix element

Making the transformationsMaking the transformations

For the slowly varying amplitudesFor the slowly varying amplitudes

:We adopt the rotating wave approximation (RWA) which amounts to neglecting :We adopt the rotating wave approximation (RWA) which amounts to neglecting

oscillating sum frequencies as opposed to those oscillating with oscillating sum frequencies as opposed to those oscillating with

Where we have used the following properties of the Laplace transforms:Where we have used the following properties of the Laplace transforms:

Which satisfy the initial conditions, as well as the normalization conditionWhich satisfy the initial conditions, as well as the normalization condition

The inverse Laplace transform ofThe inverse Laplace transform of

Precession of Bloch vector R about the effective field for

(a)

(b)

Let us introduce the atomic operatorsLet us introduce the atomic operators

Defining the atom-cavity field coupling constantDefining the atom-cavity field coupling constant we can writewe can write

In the interaction pictureIn the interaction picture

Terms, which do not conserve the total energy of the system, are dropped in Terms, which do not conserve the total energy of the system, are dropped in the rotating wave approximation,the rotating wave approximation,

The total Hamiltonian becomesThe total Hamiltonian becomes

Known as the Jaynes-Cummings modelKnown as the Jaynes-Cummings model

The only non-vanishing matrix elementsThe only non-vanishing matrix elements

If at time If at time t t = 0 the atom is in the upper state = 0 the atom is in the upper state and the field state isand the field state is

For the atomic population inversionFor the atomic population inversion

For exact resonanceFor exact resonance

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

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

No revivals, completely chaotic behavior, due to the absence of any phase relations between variousNo revivals, completely chaotic behavior, due to the absence of any phase relations between various

the inversion undergoes collapses and revivals, the time-scale depends onthe inversion undergoes collapses and revivals, the time-scale depends on

Each componentEach component Tends to drive the atom with its Rabi frequencyTends to drive the atom with its Rabi frequency