M3L13h.txt

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So far, we have discussed the monochromatic case,
but I really needed is a new result because I carried over
for the [INAUDIBLE] case was a perturbative result.
But I also wanted to show you that the perturbative result is
one limiting case of the exact solution which I trust derived
by analogy to spin one half.
We just had the result that in perturbation theory
for sufficiently short times-- we discussed all
that-- that the excited state amplitude has
to follow Independence.

So this is nothing else than-- want
to make sure you recognize it-- Rabi
oscillations at the generalized Rabi frequency.
The generalized Rabi frequency is simply the detuning
because it's a perturbative result.
In perturbation theory, you don't
get power brought in because you assume that your dry field is
perturbatively weak.
So therefore, they the Rabi oscillation
are now a Rabi oscillation where the Rabi
frequency, the generalized Rabi frequency,
is delta the detuning.
And this is just rewriting.
This is this result here.
I wasn't commenting on it, but this is
nothing else than the detuning.

I'm just reminding you what you get from perturbation theory.
Power [INAUDIBLE] is not part of perturbation theory.
OK.
So this is our perturbative result.
And now we want to integrate over
that because we have a broadband distribution of the light.
So what we have to use now is the energy density W of omega.

The energy of the electromagnetic field
is one half absolute the energy density
of the electromagnetic field is one half epsilon naught
times electric field squared.
Well, if you have many modes, we add the different modes
in [INAUDIBLE] and we still have the same relation
between the electric field squared and the total energy.
But the total energy is now an integer over d omega.
We integrate over frequency over the spectral distribution
of the light.
So this is how we go form energy density to electric fields.
But now he want to evaluate this expression and what
appears in this expression is that Rabi frequency.
Well, what we have to do now is we have to go back
from their Rabi frequency.
We assume linearly polarized light in the x direction
to the electric field.
OK.
We want to now take this expression
and sum it up over all modes, which means we integrate over
[INAUDIBLE] the Rabi frequency squared
as an electric field squared.
And the electric field squared is
obtained as an integral over the spectral distribution
of the light.
So this means we will replace the Rabi frequency
in this formula by an integral over the energy
density of the radiation.
We half the matrix element squared as a prefector.
I just tried to rederive it, but I
think the prefector is 2 epsilon naught.
So, yes.

In perturbation theory, the probability
to be in the excited state is-- let's just take
all of the prefectors.

Now, I changed the integration variable from omega
to detuning.
We just go from resonance-- we indicate relative
to the resonance.
So our energy density is now at the resonance omega
naught plus the detuning.
And we have this Rabi oscillation term.

OK.
So this is nothing else than taking our perturbative Rabi
oscillation formula, which is coherent physics--
and integrate over many modes.
I'm one step away from the final result.
If the energy density is flat, it's broadband.
So for the extreme broadband case,
we can pull it out of the integral,
and then we are left only with this function F of t.
If I plot this function versus delta,
we have something which is wiggles.
Then there is a maximum, and it has wiggles.
The width here is t to the minus 1.
And the amplitude is t squared.
And this is the excited state amplitude squared.
So if we integrate that over delta,
we get something which is linear in t-- something which
crosses t squared and has a width of 1 over t.

So the function F of t, which is under the integrant,
starts out at short times proportional
to t squared as we discussed.
Maybe my drawing should reflect that.
But then it becomes linear.
So for long times, the function F of t becomes linear in t
and the delta function in the detuning delta.
This is what you have seen many times in the derivation
of Fermi's golden rule.
I'm running out of time now.
I pick up the ball on Wednesday, and we'll discuss that result
and put it into context.
But the take home message-- and what I really wanted to show
you is-- that we do have coherent Rani oscillations.
And by just performing the integral
over this broad spectrum of the light,
we loose the Rabi oscillation, and we find rate equations,
Fermi's golden rule, and excitation probability
proportionate to a t.
And we have done the transition from coherent physics
to irreversible physics.
This is all hidden in this one formula,