M5L25c.txt

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So now we pretty much know what we have to do.
We want to use the symmetry of the [INAUDIBLE]--
we consider ground and excited state of each atom as spin 1/2,
but now we want to look at the total spin,
the total pseudo-angular momentum of the two atoms,
and later we extend it to n atoms.
So we want to use now the power of the angular momentum
description.
And that goes like follows.
We have four states of two atoms.
And this is gg, ge, eg, and ee.
And if I denote with ground states spin down,
excited states spin up, I'm talking about two
spin 1/2 states.
And two spin 1/2 states can couple to S equals 1,
total S equals 1, and total spin S equals 0.
And that's what I've done here.
I've arranged the states ee, the symmetric superradiant state,
the ground state, and the subradiant state.

I've arranged an energy level diagram.
Here we have zero excitation energy,
here we have one excitation energy,
and here we have two excitation energies of the atom.
But I've also labeled now the spin labels
for the combined system.
Those symmetrized states correspond to a spin equals 1.
It's a triplet letter with three different magnetic quantum
numbers.
M equals plus 1 means everything is highly excited.
M equals minus 1 means we are in the total ground state.
And here we have the singlet state,
which is the anti-symmetric state,
or the subrradiant state.
And our interaction Hamiltonian is the total spin plus minus.
It is the raising and the lowering operator.
And you know that the raising and lowering
operator for this speed is only making transitions
within a manifold of total S. It just changes the n quantum
number by plus minus 1.
So the Hamiltonian cannot do anything to the singlet state
because there is no other singlet state to couple.
But within the triplet manifold, the sigma plus sigma
minus operator is creating transitions
between the different n states.
And the coupling constant, which for an individual atom
was little g, is now factor of square root 2 enhanced.
And we will see in a few minutes that for n atoms,
it's square root n enhanced.
And if n is big, that's where the word super in superradiance
comes home.
Actually, let me just quickly add the diagram
for the single atom.
The single atom has only an excited state, a ground state.
It corresponds to S equals 1/2.
And we have magnetic quantum numbers of plus 1/2
and minus 1/2.
And the coupling due to the light-atom interaction
goes with a coupling constant g.
So the key message we have learned here
is that we should, when we have several atoms
within an optical wavelength, we should
use for their description symmetrized
and anti-symmetrized states.
Or when we generalize to more than two atoms,
we should use-- we should just add the total angular
momenta by treating each atom as a pseudo-spin 1/2.
And it is this angular classification
which tells us how the radiation proceeds
because the coupling to the electromagnetic field
is only involving the lowering and raising operators
for the total spin.
And this only acts on a manifold where the total spin
S is conserved.
And what we get is transitions with delta n plus minus 1.