M1L2g.txt

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There's one thing I want to do about classical spins
and then we do the full quantum treatment.
And this aspect of classic spins is
called rapid adiabatic passage.

So we want to add one more twist to our discussion.
So far we have assumed a static field and a rotating field
which rotates at one frequency.
But now we want to change the frequency of the rotating
field.
So we have our magnetic moment in a static magnetic field
which is how a quantization axis,
and now we have a rotating field,
but we increase the frequency of the rotating field from slow
to fast, and ask what happens.
And the result is, that by increasing the frequency,
sweeping the frequency through the resonance,
we can do something very useful.
We can invert the spin.
We can turn over the magnetic moment in a very robust way.
And this is the concept of rapid adiabatic passage.
A lot of you may be familiar with the concept
of a Landau-Zener transition.
What I'm telling you is exactly the classic counterpart
of the Landau-Zener transition, actually
in many cases, when it comes to spin physics,
the classical physics and the quantum physics is the same.
So that's why I want to discuss rapid adiabatic passage
and Landau-Zener physics first in the classical environment.
So rapid adiabatic passage is a technique
for inverting, turning around, spins
or magnetic moments by sweeping the frequency of your drive
field across the resonance.
So adiabatic means that this frequency sweep has to be slow.
Slow is slow compared to the Larmor frequency.

At any given moment, the magnetic moment
precesses at the Larmor frequency,
which is the gyromagnetic ratio times the effective magnetic
field.
So, it's sort of quasi-stationary, this spin
precesses around the effective magnetic field
and we want to change the frequency of the drive,
slow compared to that motion.
Well, the word rapid adiabatic passage,
has the word adiabatic, which means slow, but also rapid.

The word rapid means we have to be rapid
compared to relaxation processes, which we do not
discuss here, in an idealized environment.
For instance, if you do rapid adiabatic passage
in an environment where the atoms can collide.
Rapid means you have to do it fast enough before you have
decoherence due to collisions.
So slow compared to the Larmor frequency and rapid
compared that to all the things I'm not mentioning here.
Rapid compared to decoherence and relaxation processes.

I will not set up differentially equation
and solve them.
I'm going to give you the intuitive picture of what
goes on, but then also derive what
is sort of the criterion for adiabaticity
which we have to fulfill.
So what are our ingredients?
We have magnetic moment mu.
We have a static field, B naught.
We have a drive field, B1, which rotates at the frequency omega.
And we assume that the rotating field is
smaller than the static field.
It's not absolutely necessary but you
apply a big static fields and then you
have a perturbative drive.
That's a standard situation.
Or we always want to have a quantization axis-- which
is given-- but the z-axis is defined
by the static magnetic field, but it's only
defined by the static magnetic field
if the transverse field is not much
larger than the static field.
Otherwise we're talking about a somewhat different problem.
OK.
And let me just assume, to be specific,
we need to discuss what happens if it's not the case.
We assume that we start with the frequency omega, which
is much, much smaller than the Larmor frequency-- much,
much smaller than the resonance.

So what does it mean for our effective magnetic field.
Let me just catch it for you.
Remember, our effective magnetic field--
we have a field B naught.
We have a drive field B1, which we assume is smaller.
But then, when we go to the rotating frame,
we have a fictitious field, but if the frequency omega
is below the Larmor frequency, this fictitious field
is very small.
So therefore, we start out in a situation
where the effective field is pretty much pointing
along the z direction.
OK.
So just to remind you-- so we have
a situation where the effective field is just at a tiny angle.
And if you started out with our magnetic moment aligned in z--
and we assume B 1 is really perturbative.
Pretty much, the magnetic moment is very tightly coupled.
It has a very small precession angle.
Or if you want, if you take the magnetic field
B1 to be perturbative, you can say
the magnetic moment is aligned with the effective field.
That's the limit of-- that the cone angle of precision
is very small.
So let me write that down.
This was omega, much, much smaller
than the Larmor frequency.

So now, we want to turn up the knob
on the frequency of the drive.
We want to rotate the drive field B1 faster and faster.
And the picture you should have, is-- I just wiped it awayy--
but what that means, is the effective field,
the fictitious field has now a longer and longer comportment.
And on resonance, the fictitious magnetic field
will cancel B naught.
So in other words, at this point,
when B naught is canceled, the effective field
has only the B1 component in the x direction.
So therefore, when we go to delta equals zero,
the effective field is only B1 and points in the x direction.
So what we have done is, by changing the frequency,
by ramping up the frequency to resonance,
we have tilted the effective magnetic field
from the z direction into the x direction.

And at any given moment, I mean, we
know what the exact solution of the magnetic moment is--
At any given moment, the magnetic moment
precesses around the effective magnetic field.
And if the precession is very fast,
and the effective magnetic field is slowly rotating,
the magnetic moment is just following.
So therefore, at this point, when we are at resonance,
we have tilted the magnetic moment by 90 degree.
Just one second.
And if you go with the frequency much higher
than the Larmor frequency, then our fictitious magnetic field
is much larger than B naught.
And therefore, the effective magnetic field
points now in the minus z direction.
So, the idea is, that as long as this rotation
of the effective field from being plus z,
in the x direction, and into minus z, is slow enough.
The rapid precession is locking, is keeping
the magnetic moment aligned with the effective magnetic field.
And we have just a handle we invert.
We move around the magnetic moment.

So, in the adiabatic limit, the spin precesses tightly.
And by tightly, I mean the angle theta is small around B_eff, B
effective, and follows the direction
of the effective magnetic field.
So we are rotating an effective field.
We are rotating the magnetic moment,
but we're not rotating anything in the laboratory.
The only thing we are doing is, we
are changing the frequency of rotating magnetic field.

Questions?
Jenny?
Oh yeah, I was just thinking.
Can't you also do this by keeping the frequency the same
and just ramping up the strength?

Yes, I mean, the essence here is that
the effective magnetic field is scanned through resonance.
And what you are suggesting is, if the frequency is constant
that would mean the fictitious magnetic field is constant.
But if we were given fictitious magnetic field,
and we have a huge field B naught--
the effective magnetic field points out, points up,
but if you now make the static field, B naught,
smaller and smaller, go through resonance
and make it even smaller, we have also
done an inversion of the effective magnetic field.
The result is the same.
Actually sometimes in the laboratory,
if you have a synthesizer which is not easily
computer controlled, we do an analog sweep
of the magnetic field, so we actually
change the Larmor frequency of the atom,
instead of changing the drive frequency.
What really matters in the whole business
is the relative frequency between the two.
Is it necessarily true that the spin always precesses tightly
around the effective field?
Isn't it at resonance, the radius of the circle
in precession is equal to mu?

Before we found that the radius was mu sine theta.
And then sine theta, it was like omega r over--
Yes, but the way how we've set it up here-- now we start out--
if you are far away from resonance,
the angle theta is infinitesimally small.
Something confuses you.
Yeah.

See, what we have solved before-- We've done something
else before.
What we've discussed before is if we have a spin which
is aligned, and then we looked at the-- you can say
the transient solution-- We switched suddenly
on the rotating field, so then we've
sort of described that we have a magnetic field which
is in the z direction.
We leave it on.
Our spin is aligned in the z direction.
And if you then suddenly switch on the rotating field,
that would mean you have suddenly
created an effective field which is tilted.
So then, by the sudden switch on of your drive,
you have now an angle theta, and the magnetic moment
precesses with the precession of angle theta.
But what we are here doing is, we
are ever so slightly changing the angle theta.
And then, the spin stays aligned.
That's the difference.