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M5L25h.txt
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M5L25h.txt
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#
# File: content-mit-8-421-5x-subtitles/M5L25h.txt
#
# Captions for 8.421x module
#
# This file has 85 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
What you want to discuss now is the rate,
which is the matrix element squared, or the intensity
of the observed radiation, relative to a single particle.
So the intensity, and this is what we are talking about,
is now the square of the square root.
Which is S minus one plus M times S plus M.
Pretty much, this is the complete description
of superradiance for strongly localized atoms.
It's all in this one formula.
Once we learned how classify the states,
we can just borrow all the results
from angular momentum addition and angular momentum operators.
So, I want to use this formula for the intensity
and look at which is the most superradiant state,
the state where all the particles are symmetric.
And this is the state where the spin is N over two.
So I'm looking now at the ladder of M states
and want to figure out what happens.
So the maximum M state is M equals S.
All the atoms are excited.
And now the first photon gets emitted.
So just put S equals M equals N half
into the formula for the intensity.
And you find that the intensity gives us
this expression is just N.
So we have N excited atoms.
And they initially emit with an intensity, which is N.
And this is the same as for N completely independent atoms.
So nothing really special to write home about.
But now we should go further down the ladder.
And let's look at the state which has M equals zero.
Well, then the intensity, or the matrix element squared,
for the transition which goes to M
equals zero has an intensity which is N over two.
I mean, just look.
S is N over two.
And M is zero.
So we have the question whether we have odd or even number
of particles.
But it doesn't really matter.
What dominates is always the big factor, N over two.
So what we find out is that we have an enhancement,
huge enhancement over independent atom,
because this intensity goes with N square.
And this proportionality to N square, this
is the hallmark of superradiance.
So this is what is characteristic
for superradiance.
We have an N times enhancement relative to a single atom.
Now in the classical picture, that
should come very naturally.
If you have all the spins aligned,
and they start their Larmor precession.
There is not a lot of oscillating dipole moment.
But when half of the spins are de-excited,
they are now in the xy plane.
Now you have this giant antenna which oscillates and radiates.
So it's clear that at the beginning,
the effect is less pronounced.
And if you are halfway down the Bloch sphere,
then you would expect this N times enhancement.
But now let's go further down the ladder
and ask what happens when we arrive at the end?
So I'm asking now, what is the intensity when
the last photon gets emitted?
There is only one excitation in the system.
And the answer is, it's not one like an independent atom.
If you inspect the square root expression, you find it's N.
So we have one excitation in the system,
but it's complete symmetrized.
And therefore, we have an N times enhancement.
And I want to show you where it comes about.
So there is only one particle excited.
And here we have an N times enhancement.
By the way, they states with the classification S and M are
called the Dicke state.
And this state here, which has a single excitation but it
radiates N times faster than a single atom,
is very special Dicke state.
And there is currently an effort in Professor [? Woletage's ?]
lab to realize in a very valid controlled way,
this special Dicke state in the laboratory.
These are sort of non-classical states
because they are not behaving as you would, maybe, naturally