M3L13f.txt

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So the first discussion will show Rabi flopping.
I don't know how many times we have looked at the Rabi
oscillation, but these are now Rabi oscillation
between two electronic states covered by a laser beam.
And I want to show you how this comes about.
And when I said strong driving, well, we
have only a limited time window before spontaneous emission
happens.
We have to discuss the physics we want to discuss
in this short time window.
And if you want to excite an atom
and see Rabi oscillation in a short time,
you better have a strong laser beam.
So this is why the monochromatic excitation we discussed
will pretty much automatically be in the strong coupling
limit.
OK, so what do we have?
We have a ground and we have an excited state.
We have a matrix element.
We know now where it comes from.
And we have a monochromatic time dependence.
In perturbation theory, we-- in perturbation theory,
we build up time [INAUDIBLE] amplitude in the excited state.
Because we couple the ground state
with the off diagonal matrix element to the excited state.
And we have to integrate.
From the initial time to the final time,
we have the time dependence of the electromagnetic field
and we also need the time dependence
of the excited state.
So when I integrate now over T prime,
I take out the ground state amplitude
because we are doing perturbation theory
and we assume that for short times in leading order,
the ground state is one as prepared initially.
This integral can be been analytically.

Some of you may remember that the minus 1
has something to do with the lower bound of the integral.
And when we discuss the AC polarizability,
we said this is a transient and we neglected it
for good reasons.
But now we're really interested in the time evolution
of the system.
So now we have to keep it.
OK, we are interested in the probability in the excited
state.
So we take the above expression and square it.
And we find the well-known result
with sign square divided by omega minus omega eg.

OK, the two important-- so this is pretty much
just straightforward writing down in analytic expression.
But now let's discuss it.
For very short times, and this is an important limiting case,
the probability in the excited state
is proportional to time squared.

And this is important.
You are not getting a rate which is proportional to time.
We are obtaining something which is
time square and the proportionality
to t square means it's a fully coherent process.

So whenever somebody asks you you
switch on a strong coupling from the ground
to the excited state, what is the probability in the excited
state?
It starts out quadratically.
The linear dependence-- if I'm just
going the whole rate equation or such-- only come later.
This is a very universal feature.
And even if you use broadened light, for time window delta
t, which is shorter than the inverse bandwidths
of the light, talking about Fourier [INAUDIBLE],
you don't have time to even figure out
that your light is broad and not monochromatic.
For very short times, the Fourier limit
does not allow you to distinguish whether the light
is broad or monochromatic.
So what I just derived for you-- an initial quadratic dependence
is the universal behavior of a quantum
system at very short times.
Because it simply says the amplitude in the excited state
caused linearly in time in the probability quadratic.
OK, so this is for very short times.
But if you look at it now for longer times,
we have actually oscillatory behavior.
And these are Rabi oscillations.
But there's one caveat.
So we have derived.

However, we have derived them only perturbatively by assuming
that the ground state has always a population
close to 100%, which means we have assumed
that the probability in the excited state
is much smaller than 1.
Otherwise, we would repeat the ground state.
And this is only fulfilled if you inspect the solution.
The solution is always self consistent
if you have an off resonant case where the Rabi oscillation are
only transferring a small fraction of the [INAUDIBLE]
population or the excited state.
Of course, you all know that Rabi
oscillations-- this formula is also valid on resonance.
And you can have full Rabi flopping.
But I want to make the case here.
Distinguish carefully between monochromatic radiation
and broadband radiation.
For that, I need perturbation theory.
And therefore, I'm telling you what perturbation theory
gives us in short times and in terms of Rabi oscillations.

You're saying we assume strong coupling with respect
to the atomic linewidth, but weak coupling with respect
to the resonance, it seems?

It's simple but subtle, yes.
So what we have is we assume we switch
on a monochromatic laser.
Since we do not include spontaneous emission, which
will actually damp out Rabi oscilation--
we'll talk about that later-- we are limited here
to short times, which are shorter
than the spontaneous decay.
And now I gave you one universal thing.
At very, very short time, it's always quadratic.
It's a coherent process.
So that's one simple limiting except case
you should keep in your mind.
But now the question is, if you let the time go longer,
something will happen.
And there's several options.
One is if times go longer, spontaneous emission happens.
OK, we are invalid.
The other possibility is when time gets longer
and we are on resonance, we deplete the ground state.
Our formalism perturbation theory doesn't deal with it.
But if we are off resonant, we can allow time
to go over many Rabi periods and observe perturbative Rabi
oscillations.
So this is how we have formulated it.
We do perturbation theory of the system
without spontaneous emission.
And eventually, we violate our assumptions,
either because spontaneous emission kicks in
or because we deplete the ground state when we drive it too hard
or if you go too close to resonance.