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M3L16d.txt
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M3L16d.txt
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#
# File: content-mit-8-421-3x-subtitles/M3L16d.txt
#
# Captions for 8.421x module
#
# This file has 165 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
So we have done the work.
What I want to do now is just make those terms
into an energy level diagram.
I like sort of pictorial representations,
and each term becomes now a graphical [INAUDIBLE].
So let us assume we have a system hydrogen s 2 p state.
But let's say, generally, we go from a j equals zero to j
equals 1 state, which has three components.
Now I have set it up in such a way
that-- we need a little bit extra space-- I've set it
up in such a way that the states here, this is m equals zero,
this is m equals plus 1, and this is m equals minus 1.
So therefore-- let me just use color coding now--
so this one here is delta m equals plus 1.
So this one always moves to the right.
It changes angular momentum buy one,
so it can always move to the right.
Whereas the other one, delta m equals minus 1,
moves to the back.
Absorption is e to the i omega t, always moves up,
and stimulated emission moves down.
So with that, what happens is this term
here transfers one unit of angular momentum and energy.
So that would mean this term goes up here.
It could go up here if there were a state.
The other term-- let me use, what color do we
have left, green, let me use a green color-- is driving
the process in the opposite direction.
But now we have to also consider that you can go down here,
and you can go down to a virtual state.
A virtual state is just something
which has the same wave function as a state,
it just has an e to the i omega t,
which does not-- it's a driven system, you drive it.
You mix a state, you mix a state at the drive frequency,
and it just means, in this case, this state
has an oscillation e to the i omega t, which is very, very
different from what a state which is populated would have,
and this is what we call a virtual state.
So, in other words, what is possible is we have our three
states, plus/minus 1 and zero, but this
is the spatial wave function, including angular momentum.
But we can now drive it by plus omega and minus omega,
and therefore we can have it as virtual states
pretty much at any energy we want.
But this process here is not possible
because this would require to go to a state which has m
equals 2, which does not exist.
So now what I've shown here is, if it
would start in the m equals zero state,
I've shown you the four terms to a co-rotating
and to a counter-rotating.
If you neglect this virtual state, which
has a detuning of about 2 omega, or two resonance
frequency of the atom, this is a rotating wave approximation.
One term is responsible for absorption,
the other term is responsible for stimulated emission.
But, if I don't make the rotating wave approximation,
I've those two extra terms.
So let's now ask-- so this is only the right-handed light--
and I want to sort of play a little bit with this concept--
if I would take the left-handed light,
I would add four more arrows, two more here,
and two more here.
But let's just keep the situation
as simple as possible.
But I really like that you write down right-handed, left-handed
side, decompose it into its components,
and each component is now in this diagram connected
to an arrow, where one direction is angular momentum,
the other one is energy.
So let me, now, talk about other energy diagrams.
And this will lead to the answer, well,
can we create a situation where we have only two terms-- which
would be the simple two-level system-- can be directly
realized without any approximation,
without any rotating wave approximation,
at two-level system.
OK.
So if we had two levels, which have only m equals zero,
and m equals one-- so this would be
the situation I just discussed with those two levels.
So the only way how I can fit in this arrow is this one,
and the diagonally downward arrow is that.
So in this case, rotating wave approximation is not
an approximation, it is exact.
But some purists will actually say,
hey, you can never realize it, when you have an m
equals plus 1 state, then you always
have an m equals minus 1 state, and then
you have a virtual state down there,
and then you get two more terms, which
are the counter-rotating terms, which I just showed above.
So while I would say, if you have a neutron star which
makes an infinitely high magnetic field,
you can have a huge [INAUDIBLE] splitting
between m equals plus 1 and m equals minus 1,
and completely move one of the angular momentum
state out of the picture.
But of course, in the rotating wave approximation,
we are neglecting off-resonant terms
at 2 omega, omega being in an electronic excitation
energy, so I'm really talking about Siemens shifts
here to eliminate the other state which may be comparable
tow electronic energies.
So in principle, I can say, this is my Hilbert space,
and in this Hilbert space no rotating wave approximation
is needed, but it's maybe an artificial Hilbert space.
When I had a discussion with other people,
we came up with the possibility of some forbidden transition.
If you go from a doublet s to a doublet s state,
all you have is a spin system which has one half angular
momentum, plus one half, minus one half,
and then you realize that only way, how
you can fit in the orange arrow is
in this way, and the green arrow in this way.
So here you would have a situation where the rotating
wave approximation is exact.
But of course, it's not an electric dipole transition.
It's some sort of weaker transition,
which may be forbidden.
I have discussed the case where we have quantized
along a direction, I called it the k direction,
and the polarization of the electromagnetic field
was i and j, was perpendicular to it.
So let me now discuss the case where
we quantize along the polarization
of the electromagnetic field.
And you remember from our discussion on selection rules,
that this is pi light.
So in this case, our magnetic or electric field,
it is polarized along the i direction.
And the real cosine omega t gets decomposed into e to the plus,
e to the minus i omega t, and we know already one term is
absorption , one is emission.
And now, if I take my j equals zero to j equals one system,
I get, there is no-- pi light has a selection rule of delta m
equals zero.
So now I have an arrow which goes up,
green arrow with-- here is green, but now of course,
with linearly-polarized light, we can always
go down to a virtual state.
We have now four terms, two are rotating,
two are counter-rotating.
So therefore, the quick conclusion
of the last 10 minutes is that there is the possibility
that counter-rotating terms can be zero for sigma
plus, sigma minus light, due to angular
momentum selection rules.
But we have also learned, if you have the m plus 1 state
and there's an m minus state, if you have circularly
polarized light, and we drive a transition between two m
states, the counter-rotating term
does not come from the same set of two states, m equals 1.
It involves m equals minus 1.
So it's the other state which is, maybe degenerate, or only
slightly split by a magnetic field, which
is responsible for the counter-rotating terms.
Anyway, we have talked so much about rotating wave
approximation and those terms, I just
wanted to show you how it is modified
if you use degeneracy p states and angular momentum.
Any questions?