M3L16bb.txt

#
# File:   content-mit-8-421-3x-subtitles/M3L16bb.txt
#
# Captions for 8.421x module
#
# This file has 253 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------

What I want to discuss now is the situation
that we're not starting with an empty cavity.
We are starting with a coherent field.
We can also start with a thermal field
through the different experiments you can do.
What would we expect now?
So now the initial photon state is not the vacuum state,
but the thermal state.
If you're a microwave cavity and you heat it up little bit,
you have to cool it down to below one Kelvin.
People use either helium-3 cryostats or dilution
refrigerators, but if you warm it up a little bit
you have a few photons, microwave photons,
in the cavity.
Or, what's even more controlled, you can make the cavity ice
cold, but then you inject a few photons from your synthesizer
into it from your microwave generator,
and then you have a weak coherent field.
But thermal state or coherent state.

OK.
We would expect now Rabi oscillation.
However, the frequency for the Rabi flopping
is now proportional to n plus 1.
And we have our photon field in a superposition of Fock states.
So the effect that we have a superposition state implies now
that the Rabi oscillations have a different oscillation
frequency for the different doublets of states labeled
by n.
And that leads to a dephasing.
So, that would mean that if you would look at the probability
to be in the excited state, just think about it in we have
a wave function where the atom starts in the excited state,
and the photon field is in a superposition.
So now you have a two-component wave function
which has different parts.
And each part has a specific Rabi frequency.
So you would have oscillations.
Let's say there is a certain probability
that the cavity is in a vacuum, and then
that means there is a component which
oscillates at the vacuum Rabi oscillation frequency.
But if you have a component in your coherent
or thermal state which has two photons in it,
then you have Rabi oscillations which are faster.
And now you have to superimpose them all.
And if you all superimpose them you
find that there is a little bit of oscillation,
but then there is sort of a damping.

Eventually, if you have only a small number of photon states,
then there will be a time where you
have sort of at least a partial commensurability.
You have, maybe, five frequencies.
You know, square root of five, square root of four,
square root of three, square root of two.
But then there is sort of a time where
all these different frequencies have done an integer
number of oscillations each, and then you
get what is called a revival.
And if you go to a large photon number,
you have square root of 100, square root of 99,
square root of 88, the revival will
happen at later and later times, and eventually
at infinite times if you use a microscopic field.
But for small coherent states, or thermal states,
which only involve a few photons,
you will get a revival phenomena.
And this has indeed been observed.
This was actually the PhD thesis of Gerhard Rempe
and it shows the probability in the excited state.
They had previously observed the Rabi oscillations
at early times, but now the experiment
had to be adjusted, I think, by using slower atoms
to observe the longer time.
And here, well, 1987 for the first time revivals
have been seen.

Let me dwell on that.
First, are there any questions about what happens now?
Atoms into cavity do Rabi oscillations.

If the photon field is a superposition
of only a few states due to this pseudo commensurability,
you find times where you have revivals.

I just worked out something this morning
which I think is nice because it will
highlight how you should think about spontaneous emission.
So let me discuss.
Doesn't really matter, but I want
to give you a specific example that we have a coherent state.
A lot of you know what a coherent photon state is.
For those who don't, it doesn't really
matter for what I want to explain and recover
that in 8.422.
22.
But if you have a laser, or if you have a microwave generator,
what comes out is a field which has a normalized amplitude
of alpha, but your field is in a superposition state, or Fock
states.
With these pre-vectors, I just wanted to give you an example.
What I really just need is that we
have a coherent superposition of number states.
We've prepared that.
So now we have one at atom in the excited state.
It enters the cavity which has been prepared
with a short pulse from a laser or microwave synthesizer
in this state alpha.
And now we want to discuss-- So this is at t equals zero.
And now I want to discuss what happens as a function of time.
Well, we know that if we have one doublet
n we have Rabi oscillations between the atom
is in the excited state, and we have n photons.
Or it has emitted the photon, and then we
have n plus one photon in the cavity.

But now we have a superposition state,
and we have amplitudes a n.
This includes everything.
It includes everything a two-level atom does
in a single mode of a cavity.
And this is spontaneous emission, stimulated emission,
and reabsorbtion.
But I want to use that now to discuss with you misconceptions
about spontaneous emission.
We had discussed vacuum Rabi oscillations or Rabi
oscillations when we have n photons in the cavity.
This was our two-level system, our Hamiltonian, and all
we get is Rabi oscillations with a Rabi frequency of omega n.
And now we have to do averaging.
I'm now discussing that we have a coherent superposition
of a number of states.
Let's say a pulse of coherent radiation and coherent state,
and this is what we get.
You can now, if you want, put in a zillion other modes.
Have another sum over all the other modes you want.
So I'm just doing the first step,
in discussing with you, what will happen.
But adding more and more modes will actually not
change the structure of the answer,
and will be, of course, quantitatively a mess,
but conceptually not more complicated.
So I want you to really look at that
and realize where is the spontaneity
of spontaneous emission?
Where do you see any form of randomness
associated with spontaneous emission in this expression?
I don't see it.
This is a wave function, and this time evolution is unitary.
Everything is deterministic and, depending now
how we choose our coefficient, there's even a revival.
It's not dissipated.
If a photon is spontaneously emitted and it's gone.
We saw in the single photon a Rabi oscillation
can be reabsorbed.
We saw in a slightly more complicated situation
that there are at least partial revivals,
and it now depends on how long we wait whether revivals will
take place, or whether they will be complete revivals
or partial revivals.
We don't need a revival in a coherent evolution.
The coherent evolution can just go to a complicated wave
function.
It's still a single, coherent.
wave function fully deterministically
obtained form the Hamilton operator.
Sometimes it pops into our eyes through
a reversible oscillation or through a revival,
but we don't need that.
So let me write it down, but then explain to you something.
So it's unitary.
There is no spontaneity at all.
However, we want to retrieve the classic limit.
So if you would go to the situation
that the ever rich photon number is much smaller than 1,
then the fluctuation in the photon field
around the mean number are very small.
For coherent state, the fluctuations
are the square root of n.
And then we retrieve the limit of semi-classical Rabi
flopping with a Rabi frequency omega r.
And this is the square root of n times the single photon Rabi
frequency.
And of course, for a large number of photons,
we can always make the approximation
that we do not have to distinguish
between n and n plus 1.
So this is the ultimate limit if you
would work in the limit of large photon numbers.
So the way you should look at it is the following.
This system undergoes a unitary time evolution
to a state which is rather complicated.
But if you make the number n large,
this becomes approximately a state
where you have simply-- you know what
the ray of the semi-classical limit.
In the semi-classical limit, we have a constant laser beam
with constant electric field amplitude, E,
and then we have driven Rabi oscillations between ground
and excited state.
So therefore, I don't want to show it to you mathematically,
but in the limit of large n, can approximate
this complicated entangled wave function
by the product of Rabi oscillations
between ground and excited state times a coherent photon field.

And the correction between what I just
said and this complicated wave function is like 1 over n.
Because it's sort of a 1 over n approximation,
but we have neglected terms, which the relative importance
of them is n over n.
So therefore, there are people who will say
and who will tell you when we have
an interaction of an atom with a coherent state,
and let's just think in the number of n being large,
that n times out of n plus 1 we have a coherent state,
the atom does a Rabi oscillation.
And what it does is it just emits
a photon into the coherent field and takes it back,
like in semi-classical physics.
But in one case out of n cases, is the weight.
1 over n of the weight of the wave function is sort of fuzzy.
It's not a coherent state.
It's something much more complicated.
And if you do not keep track of this complicated nature
of the wave function, and just do some simple measurement by,
let's say, just measuring the phase
of the electromagnetic wave by projecting
onto a coherent state, then you would
find that, with the probability of n,
the system was just staying in a coherent state.
And with a probability which is one part out of n,
something else has happened, but your detector cannot capture
the entanglement of that state.
And this last part is what some people associate
with spontaneous emission.

That's my view where the spontaneity in the process is.
It's not a spontaneity in the time evolution.
It's more spontaneity if you do not
care to detect this complexity, but rather map it back
to a coherent state.
And then with a precision which is 1 over n,
you retrieve the semi-classical limit.
But the difference between the semi-classical limit
and the entangled wave function, this is what some people say
is spontaneous because it's not captured by a simple picture.

I'm actually expecting some people to disagree with me,
but this is my view,  what
I have been learning form the simple examples I've given to you.