M3L13d.txt

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Let me conclude our discussion of matrix elements
by talking about something which is experimentally
very relevant.
And this is how selection rules depend
on the polarization of light.
And I only want to discuss it for electric dipole
transitions.
So when we wrote down the coupling of the atom
to electromagnetic radiation, we had the dipole operator.
But we also had, of course, the mode
of the electromagnetic field, which was characterized
by a polarization epsilon.
So until now when I talked about selection rules,
we discussed this part.
But now we want to see how it affects polarization.
Well, the epsilon, for instance, for circular polarization--
we'll talk about linear polarization in a moment--
has this representation.

So this is the unit vector of the polarization
of the electric field when it's circularly polarized.
And now, remember, we take this vector R
and expand it in the following way.
So if we multiply, now, the operator r or the matrix
elements created by this factorial operator
by the polarization, you see that one circular polarization
projects out this component.
The other circular polarization projects this out.
And later, we talk about that linear polarization projects
that out.
So when we said that we have matrix elements
for dipole transition, which can change
angular momentum or the n-quantum number by minus 1
plus 1 and 0, this is now related
to the polarization of the light-- either the photon which
is emitted or, when we use circularly polarized light,
we can only drive this transition-- that or that.
Because the scalar product of the polarization vector
and the matrix element project out only one component
of the spherical tensor.
So if you look at the expansion above,
we realize that the left and right-handed circular light
projects now out.
The spherical tensor operator T1 plus minus 1.
And since it's circularly polarized light.
And therefore, we find this selection rule,
that delta M, the z component of the angular momentum
changes by plus/minus 1 when the circularly polarized light is
sigma plus or sigma minus-- right-handed or left-handed.
OK, so this is responsible for circular polarization.
These are selection rules for circularly polarized light.
Let me conclude by discussing the case
of linear polarization.
Well, when we ask linear polarization,
if we ask for linear polarization along x or y,
well, it's linear polarization, but we should regard it
as a linear superposition of sigma plus and sigma minus.
So otherwise, if you have the quantization axis along z,
and you use light which is polarized along x or y, the way
the light talks to the atom with symmetric operators
is that the light is a superposition of sigma plus
and sigma minus.
So we have here the light k, the propagation of the light,
was along the z-axis.

But now we want to look at the other possibility, that z,
or the quantization axis, is parallel
to the polarization of the electric field,
which would mean that the quantization axis is usually
defined by an external magnetic field.
So we're talking about the situation
that the electric field of the electromagnetic wave
is parallel to the magnetic field.
Then, with this polarization, we pick out
the spherical tensor component which
is z, which is r times Y1, 0.
And that means that this polarization of the light
induces a transition for which delta m equals 0.
And this is referred to as pi light.
So maybe, if that got confusing for you,
let me just help out with a drawing.

We have our atom here.
It is quantized by a magnetic field B.
And if we shine light on it, we have the electric field
perpendicular to the magnetic field.
So this would be x and y.
And the natural way to describe it
is by using x plus/minus i Y. And we have selection rules
where delta m is plus/minus 1.
But alternatively, we can also shine light
along this direction.
And for the electric field which is perpendicular to B,
we retrieve the previous case.
We have superpositions of the sigma plus and sigma minus.
But the new case now is that the electric field is parallel
to B.
And then we drive transitions which have delta m equal 0.
So these are sigma plus and sigma minus transitions.
And this here is what is called a [INAUDIBLE] transition.

Anyway, it's a little bit formal,
but I just wanted to present it in this context.

Questions?
Nancy.
I have one slightly basic question.
When we talk about polarization in all these matrix
elements of single photon,
these are single photon matrix elements.
And so when we talk about shining a laser,
it has a polarization, but we don't
talk about polarization for single photons-- or do we?

The question is, what is a polarization?
Do we talk about polarization of single photons or polarizations
of laser beams?
Well, I would say-- let me back up and say,
we talk about polarization of a mode of electromagnetic field.
We will always expand the electromagnetic field
into modes and the mode has a polarization.
It may happen that at some point,
a photon is emitting in a superposition of modes.
But in the most straightforward description,
we always do a mode analysis.
And often, we simplify the case by saying
that the atom interacts only with one mode
of the electromagnetic field.
And maybe in the case of spontaneous emission,
we then sum over all modes.
But for each mode, there is a specific polarization.
And it doesn't matter if this mode is filled with one atom
or with a laser beam with a classical electromagnetic
field, which corresponds to zillions of photons.
And in the lab, does it always end up
being elliptical polarization in this case, then?
Because if it's many photons, then there's
a lot of modes for each of them individually.

No, it depends.
If you have an atom, and it has one unit of angular momentum,
and it spontaneously emits a photon,
if the photon is emitted along the quantization axis,
it can only be sigma plus.
If it's emitted in the other direction,
it has to be sigma minus.
Now, if you go at strange angles, then at this angle,
you have overlapped with different modes.
And you may now find photons in the superposition
of polarizations, because we have
several modes which are connected with this direction
of emission.

I think if you write it down, it's pretty clear.
It's just sort of projection operators.
And for spontaneous emission, we sum over all modes.
But for me, I always think about,
you can always think about what a single photon does by saying,
well, if I'm getting confused about a single photon,
let me figure out what many, many, many identical photons
would be.
And that would mean, instead of a single photon
in a certain mode, I have a laser beam in this mode.
And then suddenly, I can think classically.
I know what the electric field is such.
And then you go back to what is the electric field
of a single photon, and usually make the connection.
So I think at least for the discussion
of matrix elements, transitions, angular momentum,
I don't think you ever have to distinguish
between what single photons do and what laser beams do.

But there are important aspects of single photons,
non-classical aspects, which we discuss in a short while.
Other questions?

OK, that's all I want to say about selection rules.
So with that now, we can simply take the matrix element