M3L17f.txt

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We can now repeat some of this exercise
for the broadband case.
In the broadband case, the unsaturated rate,
which is the rate for absorption and stimulated emission
following Einstein's treatment of the AB coefficient,
is used by using Einstein B coefficient
times the spectral intensity.
And now we want the same situation as before.
We want to reach saturation, and saturation
happens when this is comparable with gamma,
and it's purely a definition that we
say it should be gamma over 2, but we
are consistent with what we did before.
And if you just take this equation
and calculate what the saturation intensity is,
well gamma is nothing else than the Einstein A coefficient.
Here we have the Einstein B coefficient,
and if you take the ratio between the Einstein A and B
coefficient, the matrix element, everything
which is specific to the atom cancels out,
and the saturation intensity, or the spectral density--
it's the spectral density now for broadband--
only depends on speed of light and the transition frequency
cubed.
And it doesn't make a difference whether you
have a two-level system, which has a strong matrix element,
or weak matrix element.
I could explain it to you, now at this point,
but we want to hold the idea that there
is a difference between single-mode monochromatic and
broadband excitation, until I have discussed one more
concept, and this is the cross section.

I know this topic can get confusing
because we go from one definition to the next,
so let me just summarize what I've said so far
is, we drive an atom, we have absorption,
we have stimulated emission, and we
want to understand the phenomenon of saturation.
And based on the fact how we define saturation-- namely
that the unsaturated rate is gamma over two--
we got some nice results for the saturation intensity,
and for power broadening of a low [INAUDIBLE].
It's pretty much having a definition and running with it,
and now we want to express the same physics by using
the concept of a cross section.
You know, for the following reason.
If you can do physics, you can do atomic physics
without even thinking about a cross section,
you can just have a laser beam of a certain intensity,
and I scatter light.
But often when we scatter something--
and you may be familiar from atomic collisions--
you often want to write this scattering rate
as a density times cross section times relative velocity.
And this sort of has this intuitive feeling,
if you have a stream of particles in your accelerator,
or a stream of photons in your laser beam,
you can now hold onto the picture
that each atom in your target is a little disc.
If the particle of photons hits the disc, something happens.
If it misses the disc, nothing happens.
And the area of the disk is the cross section.
So in other words, we want to now understand
how big is the disc of the atom, which will so to speak,
cast a shadow-- which is synonymous with absorption--
when we illuminate those atoms with laser light.
So for me, a very intuitive quantity.
Anyway, so all we do is, we have already
discussed the rate of excitation, which is now
the unsaturated rate, but now we express the unsaturated rate
by the density of photons times the cross section,
and the relative velocity is the speed of light.
And from this equation, we find-- because everything
is known, we have talked about that on the last few pages--
we find that the cross section is,
and this is the result-- 6 pi lambda bar squared.
Lambda bar is the wavelength of light divided by 2 pi.
So we find, that from monochromatic radiation,
the cross section of a two-level system,
is independent of the strengths of the transition,
independent of the matrix element,
it just depends on the resonant wavelengths.

Now you would say, well, but, what
is now the difference between a strong and a weak transition?
And this is shown here.
If you take your monochromatic laser, and you scan it,
you scan it through the cross section,
when you are one resonance, you have 6 pi lambda bar squared.
And the difference between a narrow transition,
with a small Einstein A coefficient,
and a strong transition with a large Einstein A coefficient,
simply means that in one case it's narrower,
in the other case it's wider.
We talked about the phenomenon of saturation.
6 pi lambda bar squared is the cross section
in the perturbative limit, or the unsaturated cross section.
Of course, if you increase the laser power,
you saturate the transition.
The atom will have a smaller and smaller cross section.
Actually that's something important you should consider.
When you have an atom and you increase the laser power,
you scatter light, and this scattered light,
or the absorbed light, saturates.
But with the cross section, we want
to know what friction of the laser light is scattered,
and the fraction of the laser light scattered
goes to zero, because you make your laser light stronger
and stronger, and the total amount of laser lights,
which is scattered, saturates.
So in other words, you have a saturation
of this scattered light.
You have a saturation of the net transfer
of atoms into the excited state, In the limit of infinite laser
power, but since the cross section is sort of normalized
by the laser power, the cross section
has this dependence, one over one plus saturation parameter,
and goes to zero.
And that means-- and this is sort
of the language we use-- that the transition bleaches out.
If you saturate the transition, the cross section
become smaller.
So when you saturate the transition in an absorption
imaging experiment, which many of you do,
the shadow is less and less black,
exactly because the cross section is bleaching out.
But the amount of light you would scatter
you would observe in fluorescence,
is not getting less, it saturates.
This is sort of just the two flip signs of the coin.
If anybody is confused, please ask a question.

OK.

So now, in this picture, we can immediately
understand why we have differences
between monochromatic radiation and broadband radiation.
If you want to saturate a transition
with monochromatic radiation, we have our [INAUDIBLE] laser,
we absorb, with a cross section 6 pi lambda bar squared,
and we have to increase the intensity of the laser
until the excitation rate equals gamma over two.
That's our definition for saturation.
So therefore, the laser intensity
is proportional to gamma because we have to-- we have
the cross section is constant, but the product
of cross section and laser intensity
has to be equal to gamma over 2.
However, now consider the case that you
use broadband radiation.
The spectrum is completely broad.
now, if an atom has a stronger transition,
it's cross section is wider, and the atom
can sort of absorb a wider part of the incident spectrum.
So therefore, if the atom has a stronger transition,
it automatically takes-- absorbs more of your spectral profile.
And therefore, the result for the saturation--
for the spectral saturation intensity,
is independent of the matrix element,
and the strengths of the transition.

In general, if you're not in either
of the extreme cases of monochromatic light
or broadband light, what you have to do is,
you have to use this cross section
as a function of frequency, and convolve it
to a convolution with a spectrum of the incident light.

And this is exactly down here.
You take your frequency-dependent cross
section, you do the convolution with the spectrum
of the incident light, and if you
assume the incident light is spectrally very broad, you can,
you simply do, you integrate over the Laurentian line
shape of the cross section, and then you
find exactly the same result as we had two slides ago--
that the saturation intensity is independent of the strengths
of the transition.