Electric dipole transition

An electric dipole transition is the dominant effect of an interaction of an electron in an atom with the electromagnetic field.

Following reference,[1] consider an electron in an atom with quantum Hamiltonian

, interacting with a plane electromagnetic wave Write the Hamiltonian of the electron in this electromagnetic field as Treating this system by means of time-dependent perturbation theory, one finds that the most likely transitions of the electron from one state to the other occur due to the summand of

Between certain electron states the electric dipole transition rate may be zero due to one or more selection rules, particularly the angular momentum selection rule.

is defined as and describes magnetic dipole transitions.

One way of modelling and understanding the effect of light (mainly electric field) on an atom is to look at a simpler model consisting of three energy levels.

In this model, we have simplified our atom to a transition between a state of 0 angular momentum (

In order to understand the effect of the electric field on this simplified atom we are going to choose the electric field linearly polarized with the polarization axis to be parallel with the axis of the

In fact if we were to choose another axis, then we would be able to find another state that would be a linear combination of the previous states which would be parallel to the electric field bringing us back to this assumption of a linearly polarized electric field parallel with the transition axis.

Now, the main question we want to solve is what is the average force felt by the atom under this kind of light?

which represents the average force felt by the atom.

In here the brackets represent an average over all the internal states of the atom (in a quantum fashion), and the bar represent a temporal average in the classical fashion.

represents the potential due to the electric dipole of the atom.

The reason we use a two state model is that it allow us to write explicitly the dipole transition operator as

and thus we get the Then Now, the semi-classical approach means that we write the dipole moment as the polarizability of the atom times the electric field: And as such

Before progressing in the math, and trying to find a more explicit expression for the proportionality constant

That is that we have found that the potential felt by an atom in a light induced potential follows the square of the time averaged electric field.

This is important to a lot of experimental physics in cold atom physics where physicists use this fact to understand what potential is applied to the atoms using the known intensity of the laser light applied to atoms since the intensity of light is proportional itself to the square of the time averaged electric field, i.e.

We will use the density matrix formalism, and the optical Bloch equations for this.

The main idea here is that the non-diagonal density matrix elements can be written as

; and Here is where the optical Bloch equations will come in handy, they give us an equation to understand the dynamics of the density matrix.

Indeed, we have: and another term that describes the spontaneous emissions of the atom: Where

We can then repeat the process for all 4 matrix elements, but in our study we will apply a small field approximation, so that the electric field is small enough that we can uncouple the 4 equations.

This approximation is written mathematically using the Rabi frequency as: Then we can neglect

Indeed, the idea behind this is that if the atom doesn't see any light, then to a first degree approximation in

This can easily be solved by using the Euler's formula for the cosine.

and we can rewrite the previous equation as: and And coming back to our average dipole moment: Then it is clear that

Finally, we can write the potential felt by the atom in due to the electric dipole interaction as: The essential points worth discussing here are as said previously that the light intensity

of the laser produces a proportional local potential which the atoms "feel" in that region.

This implies that the potential is attractive if we have a red detuned laser (