M3L15q.txt

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OK.
We want to talk about the fully quantized Hamiltonian.

I want to give you the sort of paradigmatic example
of cavity QED, where an atom within an excited state
is in an empty cavity.
And now it can emit a photon into the mold of the cavity.
But this photon can be reabsorbed.
So this is a phenomenon of vacuum Rabi oscillations.
And so I want to set up the Hamiltonian and then
the equation to demonstrate to you about vacuum Rabi
oscillations.
And for me the vacuum Rabi oscillations
are the demonstration that spontaneous emission has
no randomness-- no spontaneity, so
to speak-- because you can observe
coherent oscillation, a coherent time
evolution of the whole system which is possible only
due to spontaneous emission.
So let's go there.
A few lectures ago, we had a semi classical Hamiltonian
where-- this is when I wanted to show you that a two level
electronic system can be mapped onto spin [? one ?] [? half ?]
system driven by magnetic field.
So this was when we only looked at the stimulated term
when we only did perturbation theory.
And in that situation we had the electronic excitation.
And then we had the [? drive ?] field
which was assumed to be purely classical like a rotating
magnetic field which drives spin up, spin down transitions
magnetically.
And we concluded that yes, if we use a laser field,
it does exactly the same to a two level
atom what a magnetic field does to spin up, spin down.
But now we are one step further.
We've quantized the electromagnetic field,
and we have spontaneous emission.
And this is something-- for reason
I've just mentioned-- you will never find in
spin up, spin down because it will
take 1,000 years for spontaneous emission to happen.
So now we want to actually go beyond this same classical
picture, which is fully analogous to the precision
and rotation of spin in a magnetic field,
and we want to add spontaneous emission.
So what we have here is the Rabi frequency.
What's the matrix element-- the dipole matrix
element times a classical electric field.
And we want to replace it now by the electric field
at the position of the atom, but we
want to use the fully quantized version of the electric field.

And it also becomes useful to look at the sigma x operator,
which actually has two matrix elements of diagonal which
connects ground excited and excited ground state.
And one of them is going from the excited to the ground
state.
So this is sort of lowering the energy to sigma minus operator,
and the other one will be a raising operator.
It raises the excitation of the atom,
and we will refer to it as sigma plus.

So the electric field is replaced
by the operator obtained from the fully quantized picture.
Here we have the [? pre vector, ?]
which is the electric field of a single photon or half a photon.
Whatever.
But it's factors of 2 or square root 2.
We have the polarization.
And now if you would take the previous result,
and would look at it-- Well, we want
to go to the Schrodinger picture.
And I mentioned that in the Schrodinger picture,
the operators are time independent,
so we cancelled the e to the i omega t term.
If you would go to the result we had last week
and would simply get rid of the e to the i omega t term,
you would now find operators a and a [? dega, ?]
but they would have factors of i in front of it.
That's the equation we had when we derived it.
Well, I prefer now to use something which looks nicer.
Just use a and a [? dega, ?] and you can obtain that by shift
the origin of time.
So you're not looking into the i omega t where t equals 0.
We wait a quarter period until e to the i omega t just gives us
factors of i which conveniently cancels see
the other factors of i.
So what I'm doing is just for convenience.
Then let me write down that this is in the Schrodinger picture.

OK.
So we want to absorb all constant by in one constant
now which is the single photon Rabi frequency.
We have the dipole matrix element of the atom.

There is a dot product with the polarization of the light.
And then we have the electric field amplitude
of a single photon-- h bar omega over 2 epsilon 0v.
So this is what appears in the coupling.
And we want to write it as h bar omega 1 over 2.
And this omega 1 is the single photon Rabi frequency.
And with that, we have now a Hamiltonian,
which is really a classic Hamiltonian written down
in the standard form.
It has the excitation energy times a sigma z matrix.
It has the single photon Rabi frequency.
The single photon Rabi frequency appears--
this is the single photon Rabi frequency.
But then the operator for the electric field
after getting rid of the i-s, is simply h plus h [? dega. ?]
H plus h [? dega. ?] And so this takes care of the photon field,
and the operator which acts on the atoms
are the raising and lowering operator sigma plus and sigma
minus.
And finally we have the Hamiltonian
which describes the photon field, which
is h bar omega times a [? dega ?]
a, the photon number operator.
Any questions?
Yes.

So we got rid of the e to the i omega terms.

Why should we not have [INAUDIBLE]?
I mean we are looking at the interaction
with an atom which is at rest at the origin.
Therefore into to the i kr is 0.
We will on;y consider the spatial dependence into the i
kr when we allow the atom to move.
As long as the atom is stationary, for convenience
we put the atom at i equals 0.
But in [? 8 ?] [? 422 ?] when we talk about light forces
and laser cooling, then it becomes essential to allow
the photon to move.
And this is actually where the recoil and the light forces
come into play.
But as long as we're not interested in light forces,
only in the internal dynamics, ground and excited state,
we can conveniently neglect all spatial dependencies.
Other questions?

So this is really a famous Hamiltonian.
And you also see how natural the definition
are of the single photon Rabi frequency.
So we have one half h bar omega with the diagonal sigma z
matrix.
This is the atomic excitation.
This is the unperturbed Hamiltonian of the atom.
This is the unperturbed Hamiltonian of the photon.
And now the two are coupled.
And the coupling is a product of an operator acting
on the photon field plus minus one photon,
and the other one is an operator acting on the atoms,
and it is plus minus an atomic excitation.
So let me just remind you of that.
The sigma plus and sigma minus operator.
The sigma plus is the atomic raising operator
which takes the ground to the excited state.
And the sigma minus operator is the atomic lowering operator
which takes the atom from the excited to the ground state.
So this is our Hamiltonian.
And the Hilbert space on which this Hamiltonian acts
is the product space of the atom direct product
with the states of the light.
Or in other words, the basis state
would be that we use for the atoms
the states which have 0 or 1 quantum of excitation--
so we use excited state or ground state.
And for the photon, we can just use the [? Fock ?] states where
the occupation number is n.
Questions about that?

So it's a very-- just look at it with some enjoyment
for a few seconds.
I mean, this is a Hamiltonian, which has just a few terms.
But what is behind it is of course
the power of all the definitions.
I mean, each symbol has so much meaning.
But in the end, by having this formalism of operators
quantized electromagnetic field, we
can write down-- we can catch many, many aspects of--
or we can pretty much fully describe how a two level
system in the [INAUDIBLE] quantized electromagnetic field
with that set of equations.

Of course, the fact is not that everything is so simple.
The fact is that we have-- by understanding the physics,
we have skillfully made definitions
which allow us to write everything down