M5L25b.txt

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# Captions for 8.421x module
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In order to understand it, we have
to look at an atom in the excited state,
an atom in the ground state.
And we want to write down the wave function
as a superposition of a symmetrized and antisymmetrized
wave function.

And I shall tell you, I'm going very slowly for two atoms.
And then once I've introduced the concept for two atoms,
with a few pen strokes, we can immediately discussing n atoms.
So all of the physics, all the understanding
what goes on in superradiance is already
displayed for two atoms.
So we want to have a superposition
of symmetric and antisymmetric wave function.
The symmetric one is a normalized wave function,
which is ge plus eg.
And we call that the superradiant wave
function, for reasons which will become clear in a moment.
And if you have a minus sign here,
the antisymmetric combination we call the subradiant wave
function.
Now, what happens is, we have to consider-- so we have
symmetrized the wave function.
Well I didn't really tell you why,
but it's always good to symmetrize.
Symmetry is, if you can use it, something good.
And the reason why I symmetrized it
is because I want to use now, I want
to look at the interaction Hamiltonian.
And if I look at the interaction Hamiltonian,
the one we have seen many, many times, but now for two atoms,
we will immediately realize that this interaction to Hamiltonian
is symmetric.
So therefore, if the Hamiltonian is symmetric,
it's really a good starting point
to have the wave function for the atoms expanded
in a symmetric basis.
And since I want to emphasize that the whole story I'm
telling you today has nothing to do
with the kind of second quantization.
It is about spontaneous emission,
but it's not involving any subtlety
of spontaneous emission, of spontaneous emission and field
quantization.
I want to sort of write down the interaction Hamiltonian, both
in a classical and a quantum mechanically way.
In the classical way, we have the dipole moment E1, D1.
We have the dipole moment D2.
And the atoms talk to the electric field
at position R and T. But since we assume,
and now you realize where the assumptions, some
of the assumptions, are important.
Since the atoms are localized to within n wavelengths,
they really talk to the same electric field.
There are no phase factors.
In about 55 minutes or so, we'll introduce phase factors
for extended samples.
But for now we don't.
And therefore, what the atoms couple with
is with a dipole moment which is the sum of the two
dipole moments.

So this is classical or semi-classical.
But, so what enters in the Hamiltonian
is only the sum of the operators for the two atoms.
And the same happens in the QED Hamiltonian.
And actually, I will get a little bit more mileage
out of the QED Hamiltonian, as you will see in a moment.
Because with the QED Hamiltonian we describe the atomic system
by-- so at first atom one-- with the raising
and lowering operator with the atoms interacting
with A and A-dagger.
And then I have to add the term where the index one and two are
exchanged.
So we are introducing here, where
it's convenient for a two-level atom, the spin notation.
Sigma plus and sigma minus are the raising
and lowering operator which flip the atom
from the ground to the excited state and vice versa.
But the important part now is-- and this
is where everything comes from in superradiance--
that the coupling involves, not the individual
spins-- little sigma plus and sigma one
and sigma 2-- it only involves the sum of the individuals,
I equals 1-2-2.
And later we extend the sum to n.
So therefore, what matters for the interaction
of the atoms with the electromagnetic field
is the sum of all the atomic spin operators.
And the sum is, of course, symmetric against exchange.
So therefore, when we are asking what
is the coupling of the state, which
I called the superradiant state, the one
where we had symmetrized eg plus ge.
Or we ask, what is the coupling of the subradiant state to,
well the state where both atoms are in the ground state?
Well, now we can use symmetry.
The left hand side is symmetric.
The operator is symmetric.
And now only the symmetric state will couple.
The antisymmetric state will not couple.
So therefore, the subradiant state, eg minus ge,
cannot decay.
That's why we called it subradiant.
Well, I think a better word would be non-radiant.
But non is definitely subradiant.
And for the matrix element, eg and ge,
we find that we have actually an enhancement of the coupling