M1L3i.txt

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I want to say a few words now about, rapid adiabatic passage.

But this time, by emphasizing the quantum mechanical aspect.
In other words, we have a clear understanding
what happens classically with a clear understanding what
happens in the adiabatic limit.
But I just want to sort of in the next 10 minutes,
use what we have already learned,
combine it with the quantum mechanical Hamiltonian
and tell you that, while when you are not fully adiabatic,
you actually have transition probabilities between the two
states.
So I want to sort of bring in certain-- bring in the concept
of transition probabilities to the case
of rapid adiabatic-- well, to the case of-- what
I want to say is, rapid adiabatic passage
when it's no longer adiabatic.
But that just means, when we sweep the frequency,
and we're not in the adiabatic limit.
OK.
So how do we describe it quantum mechanically?
We use the rotating frame now for convenience
that allows me write down exactly the same Hamiltonian
in the time independent picture.
So the Hamiltonian and has two parts, the diagonal part
and off diagonal part.
So if I show the energy as a function of E2 and delta.
Maybe I should times 2 over H. Just normalize it.
So then it becomes just the straight line
at 45 degree, Y equals X. So the unperturbed-- Hamiltonian
without drive has a level crossing, a level
crossing at detuning 0.
Then, we add to it the drive term.
So the full Hamiltonian with the addition
of the drive term plus delta, minus delta, and now
it has the coupling with the rapid frequency.
That means that on resonance, the degeneracy between the two
levels is split by the Rabi frequency.
And if I now show you the energy eigenlevels of this 2
by 2 Hamiltonian, it will asymptotically
coincide with a dashed line.
Do this and do that.
So in other words, I'm just reminding you that non-diagonal,
matrix element turns a crossing
into an avoided crossing.

So when we take the frequency, omega and we sweep
the detuning, so we change delta, and do
a sweep of the frequency, omega, at a rate omega dot,
then we sweep through the resonance.
And in one limit we have rapid adiabatic passage,
or in general, we have-- we
realize the Landau-Zener problem of a sweep
through an avoided crossing.
So what I'm formulating here is, it's
the so-called Landau-Zener crossing
or the Landau-Zener problem, which
is the quantum mechanical description of-- you
take a system by changing your external parameter, here
we sweep the frequency of the rotating field.
But by changing the external parameter,
We sweep the system through the avoided crossing.

And it has the two limiting cases,
that when we go through this crossing very, very slowly,
the adiabatic theorem tells us we
stay on one of these adiabatic solid curves.
And this is the case or rapid adiabatic passage, which we
discussed in the classic limit.
But it has also the other solution,
if you would sweep through it very, very fast
you're on the diabatic limit.
You follow the dashed line, and you start up here,
and you wind up there.

The Landau-Zener problem is actually
a problem, which is-- you find it in all text books,
but to the best of my knowledge, there
is no simple elementary derivation, which
I could give you a few minutes.
And the mathematical problem is a nice mathematical --
it's a demonstration of an exactly solvable model.
But I'm not to my knowledge explicitly deriving it.
It is not providing additional insight.
It's one of the cases where the result is
more insightful and much simpler than the derivation.
So what I want to give you is therefore simply the textbook
result. So in the adiabatic limit
we stay on the solid line.
If you do not cross the avoided crossing very, very slowly,
if you have a probability, a non adiabatic
probability to jump from one level to the other one.
And this non adiabatic probability
is expressed as an exponential function, which involves
the Landau-Zener parameter.
And the Landau-Zener parameter in this exact solution,
is omega squared times the slew rate, the omega dt or delta dt,
minus 1.
This square would go outside the bracket.
So therefore, what we find is from the exact solution.
But the Landau-Zener parameter is a quarter times--
and this should look familiar.
The omega Rabi frequency, omega Rabi frequency
squared over omega dot.

When we discussed the limit of a adiabaticity classically,
I hope you remember I gave you the argument of looking
at the adiabatic condition.
That the adiabatic case requires
omega dot to be much smaller than omega Rabi
squared.
So here, there being actually what
the P is in the quantum mechanical problem,
is just the ratio of the two quantities
we compared when we looked for the limit of adiabaticity.
So therefore, the probability for a non-adiabatic transition
is simply involving this ratio, omega Rabi
squared over omega dot.

So in other words, we know already
from the classical argument, but here you confirm it.
Adiabaticity required that this inequality is met.
Since we are using it sort of-- diabatic sweeps
in the laboratory, as long as I've
been involved in doing cold atom science,
I want to sort of go one step further and teach you
a little bit more about this formula,
and try to provide insight.
And often insight is also provided when
you apply perturbation theory.
So I know the adiabatic case is very simple.
But I want to look at the diabatic case
and then look at transition probabilities
in a perturbative way.
This is actually the way how you often transfer population
in the laboratory.
So I want to understand better the way how
we transfer population from one curve to the other one.
If you do a fast sweep, we call it diabatic.

So in other words, if you have this crossing,
and we go really fast, now what happens
is, this is the crossing between spin up and spin down.
If you go much, much faster than that Rabi frequency,
the spin has no opportunity to change its orientation.
So therefore, the wave function, the spin has to stay up or down.
And that means the system just goes straight
through the crossing.
Because spin up has positive slope.
Spin down has negative slope.
Being adiabatic, staying on this lower adiabatic curve,
would actually require the system
to go from spin up in this part of the adiabatic curve,
to spin down in the other part.
And to flip a spin, cannot be done faster than the Rabi
frequency.
So if you sweep fast.
That's what's happening.
OK.
So we have two trivial limits.
One is the adiabatic limit, we just stay on the adiabatic curve
and nothing happens.
The other limit is, the infinitely fast limit.
And nothing happens again when we
looked at the diabatic basis which is spin up
and spin down.
But now let's be almost adiabatic.
And this is a problem in which we really
want to understand physically and intuitively because that means
the system spins a small time near resonance
and there is a small probability to make a transition.

So if you are in one of you're hyperfine states,
pick your favorite atom, your favorite hyperfine states.
You rapidly sweep the frequency.
You will find unless you sweep infinitely
fast, that there's a small probability in
the other hyperfine state.
And that's what you want to calculate now.
And I want you to understand how would you
estimate and calculate the small probability.
So let's now estimate the result, namely
for this small probability, in perturbation theory.

And actually what I'm calculating for
you here is, when you do RF-induced evaporation.
I know half of the class is doing that.
If you apply an RF-field, you don't have to make it so strong
that you're on the adiabatic limit.
You are exactly in this limit.
The atom will slosh several times
through the resonance in an almost diabatic way.
But there is a finite spin for the probability,
and that's how you evaporate atoms.
And I want you now to fully understand
the derivation what is the probability of ejecting
atoms in the almost diabatic limit with RF spin flips.
And that's the limit where 90% of the BEC experiments operate.
So I hope everyone realizes it's an important question.
And also I hope everybody understands the question because now I have clicker questions for you.