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M2L12c.txt
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M2L12c.txt
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#
# File: content-mit-8-421-2x-subtitles/M2L12c.txt
#
# Captions for 8.421x module
#
# This file has 155 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
Then let's talk quickly about the third point
I wanted to discuss.
And this is the relationship to the index of refraction.
Whatever is appearing here is responsible for the index
of refraction.
The polarizability determines the dielectric constant,
and use whatever relation you want.
But for atomic physics's purpose,
this is the useful relation.
This is the index of refraction.
It's related to the polarizability,
and assuming the polarizability is small,
we can use this perturbative extension.
So roughly saying, n minus 1, the difference
of the index of refraction from the vacuum,
is proportionatal to the atomic density
times the polarizability.
OK.
And we know this is what we derive from perturbation theory.
The polarizability depends on the matrix element squared,
or 1 over detuning.
Now I want to show you that did by just using that concept
and putting in phenomenologically dissipation,
we can get the full expression of what
happens to a laser beam crossing an atomic medium,
like a Bose-Einstein condensate, how much of the laser beam
is absorbed, and what is the phase shift of the laser beam.
So in order to get interesting or simpler formula,
I want to now parametrize the matrix element
by this quantity gamma.
This will turn out to be the natural linewidth,
but right now, since we haven't talked
about spontaneous emission, just say I replace the matrix
element by gamma.
And then I can write the index of refraction
in the following way.
So the matrix element squared is now parametrized by gamma.
I have also introduced a quantity, sigma 0,
which will later be the resonant absorption
cross-section for atoms.
But again here, just use it as a parametrization.
We have not introduced any new concept.
I have just re-written this expression in this way,
involving sigma 0, which is a cross-section, and gamma, which
is an actual linewidth.
OK.
The polarizability depended-- that was perturbation
theory-- 1 over the energy denominator,
or 1 over the detuning.
Now we have used with the oscillator strings the analogy
to a classic harmonic oscillator.
And at some level, you know that every harmonic oscillator
has a little bit of damping.
And the same applies to atomic oscillators.
And you can account for the damping by putting
into the frequency denominator of the - what would turn into a Lorentzian -
1 over delta.
Give it an imaginary part.
So I'm just telling you here, yes, every oscillator
has some damping, and I can phenomenologically
account for the damping by making
the resonant frequency or the detuning
slightly complex by adding imaginary part.
So if I now express all detunings,
later on I want to express all detunings normalized
to this linewidth parameter, or normalize
for this imaginary part.
So we have the situation that this expression, gamma
over delta-- this was the expression --
the perturbative expression for the polarizability--
now requires a small imaginary part.
And that means it's a complex number,
and I can now take this expression
and separate it into an imaginary part and a real part.
So why do I do that?
Well, the index of refraction appears in the propagation
of the plane wave.
If you're a plane wave, the k vector
is no longer the k vector in vacuum.
It's multiplied with the index of refraction,
and that's what I gave you.
And now you see that immediately, an imaginary part
of the index of refraction leads to absorption,
and the real part leads to a phase shift.
OK.
So what we have now is we have this expression
for the plane wave, after it has propagated through the medium.
And we see, there is exponential absorption.
Exponential absorption on resonance,
we have an optical density that's
given by this expression.
And the second term here gives us a phase shift.
And it's clear since we have one medium which
has a certain thickness that the optical density and the phase
shift are related.
Yes.
If you are resonant, so you have the maximum optical density
and the maximum absorption, and the optical density, of course,
is 1 over detuning squared.
- a Lorentzian - whereas for large detuning, the phase shift
goes as 1 over delta.
It's the dispersive scale with detuning,
and this is also what we had when you would have started out
with the polarizability.
And not add any dissipation, because far away from resonance
you simply have the original dispersive effect,
and dissipation doesn't enter.
So anyway.
I thought I would just show that to you, because this
is a full understanding of what happens to optical beams
when they pass the atomic medium.
It's nothing else than the polarizability,
or the harmonic oscillator response
of an atom to electromagnetic variation.
The only thing which is non-trivial here is--
and that's what you will discuss later in this course--
is that in order to get the final result, which you will
want to use in analyzing your data,
you have to set the dissipative-- the damping
of the harmonic oscillator-- equal to this gamma parameter.
And that's something I cannot tell you without talking about
spontaneous emission.
So just to remind you, I introduced peak gamma just
as a parametrization of the matrix element.
Then I introduced phenomenologically little gamma
as just the damping of the harmonic oscillator.
That those two quantities for two-level system are identical.
And that's something I cannot show you here,
because this requires a discussion of spontaneous
emission, and the other modes of the electromagnetic field.
This is not covered by the discussion of the AC
polarizability.
OK, any questions?
But still-- I know I'm repeating myself--
I always find it sort of interesting
that the whole physics of absorption, the beams
are absorbed, you can pretty much
pull it out of the response of an harmonic oscillator.
It's only that the damping rate of the harmonic
oscillator-- that this is simply spontaneous emission-- that's
the only point where you have to go beyond the AC Stark shift
and bring in the quantum nature of the electromagnetic field, and all the empty modes of the vacuum.