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M5L24f.txt
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M5L24f.txt
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#
# File: content-mit-8-421-5x-subtitles/M5L24f.txt
#
# Captions for 8.421x module
#
# This file has 173 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
I've given you qualitative pictures
for lasing without inversion, and for EIT.
I want to give it a little bit more
quantitative touch not by going through to the Optical Bloch
Equations, which would be necessary to describe
all features of it, but at least I
want to give you one picture where
you can derive and discuss things
in a more quantitative way.
And this is the eigenstate picture.
So I've also done here what I've said several times,
that when we have splittings between the levels,
we can actually focus on what really matters,
namely the detunings, by absorbing the laser
frequency into the definition of our levels.
And that's what I've done here.
Instead of using levels g, f, and e,
I've have levels with g, f, e, but this
is the photon number in laser field one,
and the photon number in laser field two.
So if all the photons, the laser field one and laser field two,
would be resonance with their respective transitions,
then all those three levels would be degenerate.
But now, in the three-level system,
they are not degenerate only because we
have a relative detuning delta-- a detuning small delta
from the Raman resonance-- and we have a detuning big delta,
which is sort of the common detuning of the Raman laser
from the excited state levels.
So therefore, if I define the Rabi frequency--
so Rabi frequency is over 2 again,
coefficient, and the Rabi frequencies
are proportioned to the electric field.
Therefore, they scale with the photon numbers n
and m in the two laser beams.
So if I do that, I have a really very simple Hamiltonian.
On the diagonal, we simply have the detuning,
the detuning of both laser beams from the excited state.
Here we have the Raman detuning.
And we have two couplings to the excited state.
One is laser field one, g1, with n photons,
and the other one is laser field g2 with m photons.
Any questions about-- just sort of setting up the simple
equations which we have done a few times.
Let me focus first on the simple case
that everything is on resonance.
Then if everything is on resonance,
we have the structure which I've shown here.
You can sort of say you have three levels, which
are degenerate without any Rabi frequency
because the detunings are all 0, or all the diagonal is 0,
and then the off-diagonal matrix elements, the Rabi frequencies,
are just spreading the three levels apart.
And the general structure of this matrix
is that in the middle, you always
have a state which is just a superposition of g and f.
So it's a dark state.
It has no contribution in the excited state.
And the two outer states have equal contribution
of the excited state.
So the excited state has been distributed over the two
outer states, and the widths of those levels
is therefore gamma over 2.
And we know when we had two levels
and we were driving them with the Rabi frequency resonantly,
we had splittings which were just
given by the Rabi frequency.
And now the splitting between the outer level
and the dark state is the quadrature sum
of the two Rabi frequencies.
So that's a very general structure.
I want to go back to the situation
where one laser is a probe laser.
Let's assume this is our laser beam which wants
to go through the brick wall.
And the other laser has to prepare the system.
So let me discuss the limit where the photon number becomes
very small in laser 1, the photon number n goes to unity,
and this is much smaller than m.
So we have the situation of a weak probe field.
If we have this limit, then the dark state
has much, much more amplitude in state g.
And the state g is almost decoupled.
It's in a trivial way the dark state,
and the laser beam is very strongly mixing the state
f and the excited state.
So in this limit, we have a nice physical situation.
I should actually point out that you
can solve most of those situations
for a three-level system analytically.
It's just those expressions get long, and are not
very transparent.
So what I'm trying here is, in the classroom,
I try to pick certain examples-- weak probe field resonance--
where we can easily understand the new features which
happen in this system.
So the situation we have now prepared
is we have our dark state, which is level g.
We said we have only one photon in our probe beam.
And we have lots and lots of photons in the coupling
laser, which couples the other ground state, f, to the excited
state, e.
So we have the structure that we have
now two states which have half of the widths of the excited
state.
And we can call-- and they are both bright.
We can call one the bright state 1,
and the other the bright state 2.
And the splitting is related to the Rabi frequency.
Let me just call it delta bar.
So this is now the level structure which we have.
And what I want to now emphasize in this picture
is the phenomenon of interference.
I want to sort of show you-- I mean,
we've talked about interference of amplitudes,
but now I want to take this system
and show you how we get now interference
when we send one probe photon through the system.
What we can now formulate is a scattering problem.
We have one photon in our probe beam in a special mode.
But then, all the other modes are unoccupied.
And when we're asking does the probe photon get absorbed,
does the weak laser beam get stuck in the brick wall,
we're actually asking if it is possible
that we scatter the photon out of the mode.
And it gets absorbed, but you know absorption is actually
always a two-photon process that involves spontaneous emission.
And we have emitted the photon into another mode.
So this scattering problem is a two-photon process.
And the matrix element needs an intermediate state.
But now we have two.
We start with one photon, we have the light-atom coupling,
we can go through bright state one.
From bright state one, we have the light-atom coupling again.
And eventually, we go back to the ground state
without photon.
And here we have a detuning.
It's assumed we are halfway detuned
between the two bright states.
And then we have a second amplitude,
and it's indistinguishable.
We have a Vaidman double-slit experiment.
And everything here is the same, except that we
scatter through bright state 2.
And this matrix element, when it vanishes,
this is now the condition of electromagnetically-induced
transparency.
But we want to now understand what happens
when we detune the probe laser.
So we have set up the system with a strong control laser.
We've completely mixed the final state--
this is not final, the state f, the ground state f,
with the excited state.
And now we want to ask can a weak probe
laser go through the brick wall, or how much of the probe laser
is absorbed.
So what we want to understand now is what happens--
and this is a new feature I want to discuss now--
what happens when we detune the probe laser of delta.
Well, it's clear, and I just wanted to show you the formula,
we had two detunings here.
And if you detune the probe laser by delta,
that would mean now that those denominators are no longer
opposite but equal.
In one case, we add delta.
In the other case, we subtract delta.
And therefore, we have no longer the cancellation
of the two amplitude, and we have