M3L14c.txt

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Let me just summarize where we are in our discussion of atom
light interaction.
We've actually made a lot of progress.
We have discussed matrix elements.
We have discussed the coupling of atoms to an external field
at the level of the Schrodinger equation.
And we have done perturbation theory.
And in perturbation theory, we found Rabi oscillations,
and we found rate equations.
The feature now which is missing is, of course,
damping spontaneous emission irreversibility-- another form
of irreversibility.
But right now, our Rabi oscillations
are undamped, whether we obtain them in perturbation theory
or whether we use the spin formalism to get them even
in the resonant in the strong coupling case.
And here, for the rate equation, the way how we have solved it,
the probability to be in the excited state
just increases forever.
This system will never reach equilibrium.
So that means in both cases, we have a missing element.
And this is spontaneous emission.

So for the next hour or two, we'll
talk about aspects of spontaneous emission.
Spontaneous emission will actually, eventually,
lead to damping of Rabi oscillation
and to a saturation of the excitation.
OK, so we're now discussing spontaneous emission.
And we will discuss it in actually three levels.
One is--

I will discuss Einstein's A and B coefficient.
I sometimes hesitate.
Should I really discuss Einstein's A and B coefficient?
It's sort of old fashioned.
And I have already in perturbation theory
given you a microscopic derivation
of Einstein's B coefficient.
But everybody who is an atomic physicist
knows about Einstein's A and B coefficient.
It was really a stroke of genius to do it.
And it becomes sort of our language.
So what I'm doing here is I'm beating it to death,
but I give you sort of a short summary.
It's also sort of, I make a few comments,
which is actually amazing, that Einstein actually got results
from the A and B coefficient which you can only
get otherwise if you quantize electromagnetic field.
So it's also sort of historically interesting
that Einstein actually developed the theory of the A and B
coefficient before the Schrodinger equation,
before quantum mechanics was developed.
And often you call Schrodinger equation
the first quantization, in the field quantization
the second quantization.
So in some sense, Einstein actually
preempted or had already the results of second quantization
before first quantization was developed.
Anyway, it's a landmark paper how Einstein did it.
And that's why I want to discuss it.
But it's partially also in order to give you
the historical context.
But then of course, we want to use
the modern formalism of use of quantization
of the electromagnetic field.
And we have already obtained, just now,
the result for Einstein's B coefficient
by just looking at the induced by the absorption
rate or the stimulated rate.
But then eventually by having a quantization
of the electromagnetic field, we can also
do now microscopic, fully quantum first principle
calculation of the A coefficient.
So then if we have all ready the B coefficient,
we get the A coefficient out of a microscopic calculation.
So we don't really need Einstein's treatment of A and B
at this point.
But it's nice to see the connections.
So anyway, so this is the agenda.
Einstein's A and B coefficient to pull out
spontaneous emission without even putting it in.
Then we'll talk about field quantization which
automatically leads us to a treatment
of spontaneous emission.
Any questions?

So how was Einstein able to show that there
is spontaneous emission without sort of knowing the quantum
character of fields?
Well, the point was he knew and understood that there
would be thermal equilibrium.

He said, I know what thermal equilibrium is.
Thermal equilibrium is a Boltzmann coefficient,
a Boltzmann probability for an atom
to be in the excited state.
The probability to be in the excited state
is just a Boltzmann factor, and depends on temperature
in the usual way.
And also knew that the spectrum of light would follow
a Planck distribution.
And if you put those things together,
you go beyond that, because you are in thermal equilibrium.
This here, what we derived so far,
does not have thermal equilibrium.
And thermal equilibrium only comes through the damping
of spontaneous emission.
So therefore, by Einstein just using Boltzmann distribution
and Planck's law, he got spontaneous emission.
And this is what I just want to show you.
Or for most of you, it's a reminder.