M1L2f.txt

#
# File:   content-mit-8-421-1x-subtitles/M1L2f.txt
#
# Captions for 8.421x module
#
# This file has 207 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------

So because it's an exact result and I like the result,
Rabi flopping at the generalized Rabi frequency,
I want to derive it for you.
So I want to figure out, what is the dynamic of a spin which
is originally aligned and now it undergoes-- it
is driven by a rotating field.
Remember, the resonant case was very simple.
The spin was just doing Rabi flopping.
It was fully inverted, came back, and just
did this at the Rabi frequency.
Now we know that in the off resonant case,
there will be an effective magnetic field
and it will precess at a faster frequency, which is
the generalized Rabi frequency.
But since the effective magnetic field is not transverse--
it has a z component-- the spin will never fully invert.
So geometrically, it's very easy.
We start out with a magnetic moment at 0 time.
And I can immediately draw to you the complete solution.
The complete solution is that in the rotating frame,
this sort of precesses around the effective magnetic field.
This is the solution.
What I just want to do is-- because it takes
me three or four minutes-- I want to read from this graph,
from this drawing, one or two trigonometric identities,
and derive for you the explicit expression,
what is the value of the magnetic moment
as a function of time?
But it's clear from that, it will have a maximum value.
It precesses around the tilt direction.
And when it's over there, it is a minimum value.
But it will never completely invert.
Well, quantum mechanically, if you drive a system not on
resonance, you cannot completely invert the population.
But we'll come to that later.
So what do I need?
Well, the spin is moving here on a circle.
So let's say the spin was here at one time.
At another time, it is there.
It has moved an angle phi.
The tilt angle between the spin and the magnetic field
is what I call theta.
And the angle between the initial magnetic moment
and the magnetic moment at time, t, is what I call alpha.
These are the three relevant angles.
The tip of the magnetic moment goes in a circle.
And the circle has a radius, which is mu times sine theta.
Sine theta is nothing else than the rotating magnetic field
over the effective magnetic field, which
is nothing else than the resonant Rabi frequency divided
by the generalized Rabi frequency.
OK.
I said what I want to determine is
the magnetic moment in the z direction
as a function of time.
And for that, I defined the angle cosine alpha.
The way how I derive it, the easiest way , it's geometry.
It's three dimension and triangles and all of that.
But the best way how I can describe it for you,
let me introduce this auxiliary line, which connects the tip
of the magnetic moment at time t equals 0 and at time t.
And I call this length of this line A.
And I'm want have now two triangles where one side is A,
determine A in two different ways, combine the equation,
and we are done.
So the first way is that we have the magnetic moment at time 0,
the magnetic moment at time t.
We said the angle is alpha.
And the two tips are connected by a.
And you know in an arbitrary triangle
you have this would be Pythagoras
but in the general case, this is valid for general triangle.
So applying that to this triangle, we have a square.
b square is mu square. c square is mu square.
So we get 2 mu square.
And then this term, 2 ab is 2 mu times mu times cosine alpha.
So therefore, this is 1 minus cosine alpha.
So we've taken care of the first triangle.
I hope the drawing is not completely confusing
at this point, but why don't we just look at the drawing,
looking down the effective field.
We look down at the effective field
and then we see the circle the magnetic moment precesses
and the radius of the circle, I've already given to you.
So now we want to look down the effective magnetic moment.
We see the circle.
You want to lock down the effective magnetic field.
We see the circle where the magnetic moment precesses.
It has precessed from here to there.
We connect this line.
This was our A.
And the angle at which the magnetic moment has precesses
is phi.
And the radius, as we derived before,
was mu times sine theta.
So now just using the same equation for this triangle,
we find that A square is 2 mu square,
sine square theta, times 1 minus cosine phi.
And for cosine phi, I want to use the trig identity
and express it by half the angle.
All right.
Now we are done.
We are pretty much looking at this-- the drawing is clear,
precession around the tilt angle,
and we are just doing geometry here.
And we have now two expressions for A square.
We can set the two expressions equal.

And solve for the unknown, which is cosine alpha.
And with that, we find that cosine alpha
is 1 minus 2 sine square theta sine square phi over 2.
And the purpose of this exercise was that cosine alpha tells us
the tilt angle of the magnetic moment away
from the vertical axis.
So therefore, we have done what we wanted to do.

We know the z component of the magnetic moment as a function
of time is this times 1 minus
Rabi frequency squared generalized Rabi frequency
squared times sine squared.
And now we know the precession phi, the precession at which
the tip of the magnetic moment moves
in a circle-- we discussed it already before
and you gave the correct answer with the clicker--
is the generalized Rabi frequency, omega Rabi.
So that's a nice formula.
But before we lean back and look at it,
let me just do one tiny step.
Based on our quantum mechanical background,
we can now define the probability
that the spin has been flipped, is
the relative difference in the z component
appropriately normalized.
This is just the normalized change in the z component.
And if I called that a probability that well,
which is on the left hand side, then I
find that the probability for this classical expression for,
which expresses how much the z component
of the magnetic moment has changed.

And this is exactly this celebrated formula for spin
flips in the spin one half system.

So we have derived exactly the solution for the motion,
for the precession of a magnetic moment,
in a magnetic field plus a rotating magnetic field.
And what we found is, we found that the magnetic moment
precesses at the generalized Rabi frequency.
And, as I will show you next week,
this result is exactly the same as
for quantum mechanical expectation values.
Question?
So what if we have magnetic field oscillating just in one direction
instead of rotating magnetic fields?
Oh, that's a much more complicated problem.
What happens when we have the magnetic field, which
is linearly polarized, which is only
oscillating in one direction?
Well, light or a vector, which is oscillating linearly
in one direction, can be regarded
as a superposition of a left and right rotating field.
In other words, if you superimpose
left handed and right handed circularly polarized light,
the sum of the two is linearly polarized light.
So now we have actually the situation
that linearly polarized magnetic field-- that's
what we usually do in the lab.
I mean, we have coils.
We connect them to a synthesizer.
And the field is not going in a circle.
It's going back and fourth.
It's linearly polarized.
This corresponds to a magnetic field,
which corresponds to two magnetic field, one rotates
left and one rotates right.
But the problem is, if you now do
a transformation in the rotating frame,
do we want to rotate omega to the left or omega to the right?
So what we can do is we can pick our rotating frame.
And we are now in the rotating frame.
One of the rotating fields has become time independent.
The other rotates now at 2 omega.

And at that point, we need the celebrated rotating wave
approximation that we keep the one term we have rectified,
and the other one at 2 omega rotates
so rapidly that we say these rapid oscillations do nothing,
and we discard it.
We'll discuss it later in this course.

But the gist is if you have linearly
polarized light, linearly polarized magnetic fields,
we usually have to do an additional approximation,
the rotating wave approximation.
And since the rotating wave approximation is done always
in almost any treatment, any paper you can find,
we think it's always necessary.
But what I've shown to you is when we have a rotating field,
we don't need any approximation.
The transformation of the rotating frame is exact.

But that's the beauty of it, that when
we assume rotating frames, we can hold onto exact solutions
for longer, and only later then discuss
what happens when we introduce linearly
polarized magnetic fields.