M1L3a.txt

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I mentioned to you and explained it to you
that rapid adiabatic passage is a powerful way
to manipulate classical and quantum system.
And what we discussed is that when
a spin points in the up direction
and you sweep the resonance of an oscillating magnetic field--
the frequency of an oscillating magnetic field
through the resonance, you create
an effective magnetic field in the moving
frame which will rotate.
And the atom, when the change is done adiabatically,
will follow the rotation and therefore invert spin.
So it's a perfect, very robust method
to invert population in spin systems.

But where I want to pick up today
is the question, how slow is adiabatic?
We have to fulfill an adiabatic condition.
And we have already an idea of what
the adiabatic condition is, but now we want to derive it.
That we had this picture of a spin
which is rapidly precessing.
It always precesses around the direction
of the effective magnetic field, and the condition
for adiabaticity is that the rotation
of the effective magnetic field has
to be much slower than the precession.
Let me just make it clear by a counterexample.
If the atom precesses around the magnetic field
and the magnetic field would suddenly jump,
then the atom would now start precessing
about the new direction of the magnetic field,
and it would have completely changed its angle
relative to the magnetic field.
It would have lost its alignment with the magnetic field.
So you clearly see that the condition
is the direction of the magnetic field must not jump.
And the only other time scale is the frequency
of the Larmor precession.
So our condition for adiabaticity
is the rotation of the effective magnetic field
has to be slow compared to the precession frequency.

OK.
And so we want to now derive from that condition
the conditions for adiabaticity.
And just as an outlook to make it interesting,
what I will derive for you in the classical picture
is now something which we will later encounter
as the Landau-Zener parameter.
But Landau-Zener sweeps-- we talk about it later today.
It's the quantum mechanical version
of rapid adiabatic passage.
But we now get classically a result
which will feature the Landau-Zener parameter.

OK.
So with that, let us write down what we want to look at.
It is that the Larmor frequency omega
L, which is given by the effective magnetic field,
has to be much larger than theta dot.
Now, things in general are rather complicated.
If you are far away from resonance,
you change the frequency, but the effective field
is not changing a lot.
The critical moment is really when we have the real field,
we add the fictitious field, and that causes a rotation.
The critical moment is when we are near resonance.
So in other words, we have to fulfill an inequality.
The left side has to be larger than the right side.
But the left side is actually smallest
in the vicinity of the resonance,
and the angle theta dot is actually
largest near resonance.
That's when it sort of quickly goes through 90 degrees.

So therefore, if you want to find
the condition of adiabaticity, we
can derive it by looking at the region around the resonance.
OK.
So the effective magnetic field is the real field
minus the fictitious field caused by the transformation
into the rotating frame.
So we have the magnetic field at an angle theta with respect
to the z-axis.
I've just written down the z component for you.
And the transverse component is-- so this is this component,
and the transverse component is the amplitude
of our drive field B1.
So we can just read it from the diagram.

The resonance theta is 90 degrees.
And the correction angle is whatever
we have of the effective z field over B1.
And that means that the derivative, the angle theta
dot, the angular velocity at which the magnetic field
rotates, there's a time derivative
because we sweep the frequency.
So therefore, theta dot is nothing else
than omega dot, the sweep rate of the frequency,
divided by gamma B1.
But gamma B1 is nothing else than the Rabi frequency.
And on resonance, the Larmor frequency
is just the Rabi frequency.
Because on resonance-- sorry for repeating myself--
the fictitious field has canceled the bias field,
and the only field left is the rotating field.
But the rotating field is the Rabi frequency
with a gamma factor.
So therefore, we have the adiabatic condition
that omega dot over omega Rabi has
to be smaller than omega Rabi.
Or to say it inverse, the change delta omega of your drive
frequency, the change delta omega in one Rabi period
has to be smaller than the Rabi frequency.

So omega has units of frequency.
Omega dot is a derivative, has units of frequency squared.
And this has to be smaller than the Rabi frequency squared.

You'll find that actually quite often if you
do an adiabatic change of your trap frequency,
things are adiabatic as long as the change of the trap
frequency in one period of the trap frequency
is smaller than the trap frequency.
And you find something else, that the derivative
of your trap frequency has to be smaller than the trap
frequency squared.
So these are adiabatic conditions when you tighten up,
magnetic or optical confinement for atoms.
So this is very, very generic.
This new rate of the frequency you change
has to be smaller than the relevant frequency squared.

As I said, we'll come back to that
when we do the quantized treatment
of rapid adiabatic passage and we encounter that combination in the Landau-Zener parameter.