M5L25n.txt

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You can formally describe that.
You can now define new Dicke states with respect
to this-- to the preferred mode.
And the preferred mode is the mode in the x direction.
So what I've done is-- remember, we have those atoms.
Those atoms are now sitting at different positions, x1 and x2.
And if I define Dicke states, which
have phase factors e to the ikx1, e to the ikx2,
if now this atom excites-- emits a photon
and this atom emits a photon, well, the second photon
is x2 minus x1 ahead of this photon,
if you think of those atoms sitting aligned in a string.
But the phase factor is exactly cancelling the propagation
phase for the first atom in such a way that if you are now
coupling this stage to the electromagnetic field,
the phase factors of the electromagnetic field
in the mode cancel with those phase factors,
and you again have the situation that each excited--
each state here has an equal amplitude for emission.
So now you have n possible contributions,
and the normalization is 1 over square root n,
and everything falls into place.
And you can define that for two excited atoms with two phase
factors, and so on.
So you can use immediately the same formalism.
And what happens is those phase factors are chosen
that for the interaction Hamiltonian--
and our interaction Hamiltonian is now different, it is di.
And now, in an extended sample, we
have to keep track of the position of the atom.
So for the coupling to the mode in the x direction,
we have those phase factors.

So all the phase factors cancel.
And actually, I'm not telling you
whether this is plus or minus in order to cancel it.
You pick the sign that they all cancel.
And then you have superradiance, you
have fully constructive interference.

So all this looks now the same as superradiance,
but there are also things which are different.
And this is the following.
If the atom would emit now photons
not in the preferred mode, then remember,
we had sort of the Dicke ladder.
We had sort of this most superradiant ladder, little bit
less superradiant ladder, and eventually we
had the subradiant ladder in order of smaller and smaller
total quantum number s.
Emission which is not emission in the preferred mode
stays in each ladder, and we have the superradiant cascade.
But emission into other modes is now coupling different s states
because the operator, or the phase factor e to the ikr
has broken the compete permutation
symmetry between the sides.
We've changed the symmetry.
We've not the completely symmetric sum.
We have a symmetric summation with phase factors.
So therefore, the phenomenon is somewhat different,
but we still have superradiant cascade for the preferred mode.
And the result is that we have an enhancement
for the most symmetric for the superradiant
states, which is given by that.
And this is nothing else than the resonant optical density
of your sample.
So in experiments, many of them go on in [INAUDIBLE] lab,
where he uses collective spin and the storage
of single photon in n atoms, the figure of merit of his samples
is always the optical density, the number