M3L14a.txt

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So let's now look at the situation of broadband light,
because later today I want to discuss with you
Einstein's A and B coefficient, a very classical topic
of atomic physics, a very famous concept introduced by Einstein.
And actually, I want to use the perturbation theory
for broadband light, which we have now formulated,
to derive for you the B coefficient of Einstein's A
and B coefficient theory.
OK.
So for large time, we can now talk
about rate, which is the probability which
increases linearly with time per unit time.
And this was the matrix element squared,
2 epsilon 0 h bar squared 2 pi.
The 2 cancels, and then we have from the delta function
the spectral density at 0 detuning, which
is the spectral density at the resonance frequency.
So we have a rate equation now, that the rate equation
is the B coefficient-- Einstein's famous B
coefficient-- times the spectral density.
And the B coefficient is now the proportionality
constant in the equation above, which is pi d squared.
But now, in all the formula for the B coefficient,
there's a factor of 3, because the assumption is made
that we have isotropy of space.
The atoms are randomly oriented, and therefore dx
squared for a given polarization.
The dipole moment projected on the polarization of the light,
which is dx squared, is just 1/3 of the absolute value
of the dipole moment squared.
So in other words, just to remind you
what I have actually discussed is nothing else
than Fermi's Golden Rule.
And I could have reminded you of Fermi's Golden
Rule, where the rate is given.
I just use the standard notation of textbooks.
You take the matrix element squared.
You multiply by 2 pi.
And then you have a delta function.
And the delta function implies-- a delta function is always
a reminder that it needs integration.
So whenever you have a delta function in Fermi's Golden
Rule, you have to integrate.
And there are two possibilities.
You have to integrate over the spectrum of external fields.
That's what we just did.
The other possibility is-- which doesn't apply to what we just
discussed-- that you have to integrate over
a continuum of final states.
This will be important when we use Fermi's Golden Rule
expression to talk about spontaneous emission
where we have a continuum of final states.
So anyway, I could have just said,
let's start with Fermi's Golden Rule,
and let's jump to the final result.
But I really wanted to emphasize here the intimate connection
between Rabi oscillation, the t squared dependence,
and how this turns into a rate equation.