M5L25f.txt

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Before I start with the angular momentum formalism,
I want to emphasize that what are the ingredients here?
Well, we're talking about coherence,
coherent radiation, coherence between atoms,
and we talk about radiation.
And the important part here is the following--
that when we talk about radiation,
we have the situation that all atoms interact
with the common radiation field.
In other words, all of the spins, all of the atoms,
have to emit their photons into the same mode
of the electromagnetic field.
And therefore, you may be right in some limit
that the atoms are independent, but not the photons they emit.
They go into the same mode.
And therefore, the emitted photons cannot be treated
independently.

And that's why the classical picture
is so powerful for that, because in the classical picture,
we do a coherent summation of the field amplitudes,
so we have constructive interference, the superposition
principle of field amplitudes, built into our equations,
and deeply engraved in our brains.
And that's why, when we use classical arguments,
we automatically account for that the photons sort
of interfere, that the photons are
emitted into the same mode of the electromagnetic field.
And eventually, this leads to the phenomenon
that we have coherence and enhancement,
when we look at spontaneous emission for N atoms,
which are sufficiently localized.
So let me also discuss what we have assumed here.
Number one is, we have assumed we
have the localization of this ember smaller
than the optical wavelengths.
The other thing-- and this is really important.
We're talking here about a collective phenomenon where
N atoms act together and do something
in this-- they developed the phenomenon of superradiance,
it became much, much faster than any individual atom
could do by itself.
But nevertheless, we have not assumed,
or we have actually excluded in our description,
that there's any direct interaction between the atoms.
The atoms have no [INAUDIBLE] interaction,
they're not forming molecules, they're not
part of a solid with sort of shared electrons.
So the atoms are, in that sense, non- interacting,
and therefore in a way, as long as they are just atoms,
independent.

Finally-- and I want you to think about it--
you can think about it already for two atoms,
before we generalize it to N atoms.
Think about it-- what was really the assumption about the atoms?
Do the atoms have to be bosons?
To be in the symmetric state?

Can they be fermions?
Or can they be even distinguishable particles?

If the two atoms where one would be a sodium atom,
and one would be a rubidium atom, but let's just
say we live in a world where sodium and rubidium
atoms emit exactly the same color of light.
What we have then [INAUDIBLE] in superradiant state,
for two atoms-- one of which is sodium, and one of which
is rubidium.

Yes?
Isn't the [INAUDIBLE] assumption that the photons have
to be indistinguishable, so in respect
to, say, the [INAUDIBLE]

Yes.
It is really-- and that confuses many people.
It is the indistinguishability of the photons
they have emitted.
It is the common mode where the photons are emitted.
The atoms can be distinguished.
I mean also, we've made the assumption
that the atoms are localized to within an area, which
is much smaller than lambda.
But you could imagine you have a solid state matrix,
and you have one atom here, one atom there.
And you can go with a microscope and distinguish them.
So therefore, the moment you can distinguish them
because they're pinned down in a lattice,
or if you don't like a lattice, take two microscopic ion traps
a few nanometers apart, and you tightly hold onto two ions.
It doesn't matter whether they're bosons or fermions,
it only matters whether you have bosons or fermions when
the atomic ray functions overlap and you have to [INAUDIBLE] it.
As long as have two atoms which are spatially separated,
it doesn't matter whether they're bosons or fermions.
And that also means they can be completely different atoms.
You can already call it boson a, boson b.
Now you can call it sodium and rubidium,
and they can have different numbers of nucleus,
they can have different numbers of neutrons in their nucleus.
It could be different isotopes of the same atom.
The whole collective phenomenon comes when they
emit a photon to the same mode.