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M3L17e.txt
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M3L17e.txt
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#
# File: content-mit-8-421-3x-subtitles/M3L17e.txt
#
# Captions for 8.421x module
#
# This file has 124 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
We now want to specialize it to the situation which we often
call it , namely monochromatic radiation.
And for monochromatic radiation, the unsaturated rate
follows-- I affected something out here--
it follows the normalized line shape, which is in a range
here.
And therefore, our unsaturated rate
is proportioned to the laser power,
but I usually like to express as a power four Rabi frequency,
or the Rabi frequency squared.
So our unsaturated rate follows this Lorentzian.
And on the resonance, this part is 1.
Our rate is omega rv squared over gamma.
And the definition for the situation parameter of 1,
or for the situation intensity is that the unsaturated rate
has to be gamma over 2.
So therefore, by omega rv squared
over gamma is the unsaturated rate,
it should be gamma over 2 for saturation,
for saturation parameter of 1.
So therefore, our saturation parameter on resonance
is given by this expression.
If you set the previous result, and apply it
to this unsaturated rate, you find a saturated rate,
which shows now the new phenomenon of power
Let me illustrate it in two ways.
The saturated rate involves the saturation parameter
and the unsaturated rate, as in the Lorentzian
but this intervention appears now
in enumerator and the denominator.
So it appears twice.
But with a one step manipulation,
you can transform it into a single Lorentzian
but this single Lorentzian is now power broadened.
It no longer has the roots of gamma,
or the natural line with gamma.
It has an additional term, and this is power broadening.
The equations are trivial.
It's going to just substituting one, and getting
from an expression, simplifying it to zero Lorentzian
to a single Lorentzian so I just want to emphasize it is hard,
so we have-- if you try the transition,
we have to now the roots of the Lorentzian is now gamma over 2.
If you have no saturation, but then
if you crank up the saturation parameter,
the roots increases with the square root of the power.
That's an important insight.
The square root of the power leads to broadening.
Now let me give you a pictorial description
of what we have done here.
If you start with a Lorentzian, and we increase the power,
you sort of want to drive the system
with a stronger Lorentzian.
But we know we have a scene, which is saturation.
And of course, when you drive it stronger,
you reach the ceiling on resonance
earlier then you reach this ceiling when you transfer it
away from resonance.
So therefore, if you start with a red curve,
crank up the power, you will get moreover affect or moreover
results in the links, because you're not yet saturated there.
And this graphic or constructions,
which I've just, sort of, indicated to you,
lead now to a curve, which is broadened,
broader than the original Lorentzian.
And this is the reason behind power broadening.
I want to mention one thing here.
For the classroom discussion, I have
assumed that the light atom interaction can be described
by Fermi's golden rule, which we know is a limitation, when we
are driving-- when the system is, in effect, incoherent,
and no longer coherent, with a long discussion about the Rabi
oscillation, Fermi's golden rule, in the last two weeks.
But what I'm doing is mathematically correct.
The optical block equation, which
you will use in your homework assignment,
really include the transition from Rabi oscillation
towards Fermi's golden rule, and I'm just considering this one
limited case.
I've talked about saturation of a transition.
I've mentioned that we have defined the saturation part,
I mean, as such, that when the saturation parameter of one,
we, sort of, get into the non-denare regime
where saturation happens.
And of course, for an experimentalist,
the next question is, what intensity does that happen?
This is summarized in those equations.
It's the simplest possible algebra.
You just combine two equations.
I don't want to do it here.
We have a result for the saturation intensity,
which has two features, which I want to point out.
One is its case with omega cubed.
So the higher the frequency of your transition is,
the harder it is to separate.
Of course, it has something to do with that saturation.
You have an unsaturated rate, which
is one half of the spontaneous emission rate,
and you'll remember that the spontaneous emission rate was
proportion to omega cubed, so that's why we have,
again, the omega cubed factor.
And in addition, you can write the results in several ways.
If you've have an intensity, you can go back to photons,
you get factors of omega.
So then I said, omega cubed comes from the natural line
widths.
Yes, it does.
But it's not the only omega factor.
You can write the result actually
that the omega gamma squared dependence,
because one gamma comes from the matrix elements squared,
and one comes because you need two competing excitation
for spontaneous emission.
So anyway, this is, sort of, the result.
And you can calculate for your favorite atom.
And for alkaline atoms, we usually
find that the saturation intensity is if you really
work past, and you need them.