M3L15t.txt

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If we only consider one mode-- here, of course,
in general, the general Hamiltonian
has to be summed over all modes, and then
we would get spontaneous emission and everything
we want.
But if you have a situation, where you only
look at the one single mode, then you have what
is called the famous Jaynes-Cummings model.

And a very important result of this Jaynes-Cummings model
are the vacuum Rabi oscillations,
which I want to discuss now.
It's called Jaynes-Cummings model,
so let me describe ti you why it is a model.
Well, it assumes a two-level system,
which we find a lot of candidates
among the atoms we want.
Sure, our electrons have hyperfine states,
but we can always select a situation where essentially we
only couple two states.
We can prepare initial state by optical pumping,
and then you circle the polarized light
on the cycling transition.
And this is how we prepare in the laboratory
a two-level system.
So that's one assumption of this model.
We have a two-level system.
But the second assumption is that the atom only
interacts with a single moment, and that
requires a little bit of engineering,
because it means we need a cavity.
So let me just set up the system.
So our laboratory is a big box of volume
v. And this is where we may be quantized
electromagnetic field to calculate spontaneous emission.
And our atom here may actually decay
with a rate gamma, which is given
by the Einstein a coefficient.
And in order to describe this spontaneous emission,
we quantize the electromagnetic field in the large volume v.
But now, we have a cavity with two mirrors,
and those two mirrors define one mode
of the electromagnetic field, which will be in resonance,
only resonance, with the atom.

There will be some losses out of the cavity, which eventually
couple the electromagnetic mode inside the cavity
to the other modes in the speak of volume v.
And this is described by a cavity damping constant copper.
What is also important is when we use cavity to single out one
mode of the electromagnetic field,
the cavity volume is V prime.
And we often make it very small, by putting the atoms
in a cavity, where the mirror spacing is extremely small.
We know, and I'm not writing it down again, what the Einstein A
coefficient is.
The Rabi frequency, the single photon Rabi frequency,
which couples the atom to the one mode of the cavity,
has this important prefactor, which
was or is the electric field of one photon in the cavity.
And importantly, it involves the electric field
of the photon in the cavity volume, which is V prime.

Now, you see what our experimental handle is.
If you make this volume very small,
then we can enter this strong coupling regime, where
the single photon Rabi frequency for this one mode
selected by the cavity, becomes much larger
than the spontaneous emission into all the many other modes.
So the interaction with one mode,
due to the cavity and the smallness of the volume,
is sort of outperforming all of these many, many modes
of the surroundings.
And that would mean that an atom in an excited state
is more likely to emit into the mode between the two
cavity mirrors, than to any other modes to the side.
Secondly, of course, when the photon
has been emitted into the cavity,
the photon can still couple to the other modes
by cavity losses copper.
And now, we assume that we have such high reflectivity mirrors
that copper is smaller than the single photon Rabi frequency.
And this is called the strong coupling regime of cavity QED.
So then, we can at least observe,
for a limited time, the interplay
between a single mode of the cavity in a two-level system.
And this is the Jaynes-Cummings model.
So in that situation, the Hamiltonian,
the fully quantized Hamiltonian, the QED Hamiltonian couples
only pairs of states, which we label
those states the manifold n.
So we have an excited state with n photons,
and it is coupled to the ground state with one more photon.
Our Hamiltonian has two coupling terms.
You'll remember the other two where neglected in the rotating
wave approximation.
And we can go from left to right with sigma minus a [? deca ?]
plus.
And we can go from right to left with the operator sigma
plus and the annihilation operator a,
as long as we [INAUDIBLE] small, the rotating wave approximation
is excellent.
So let me just conclude by writing down
the Hamiltonian for the situation I just discussed.
So if this is energy, we have two levels.
The excited state with n photons, the ground state
with n plus 1 photons.
If the photons are on resonance, the two levels are degenerate.
But if you have a detune in delta,
the two level are split by delta.
And what you're doing right now is for the-- maybe just
write down the Hamiltonian-- we shift the origin.
So the zero of the energy is just halfway
between those two states.
That's natural.
So this just offsets in our equations.
And so, our Hamiltonian has now the splitting
of plus minus delta over 2.
The coupling has a prefactor, which is the single photon Rabi
frequency.
And then, the a and a [? deca ?] terms
depend on n square or n plus 1.
So what I wrote down, I will say Hamiltonian rotating wave
approximation, which describes only one pair of states.
But we have sort of, of course, in our Hilbert space,
one pair of states for each label n.
But each of them is sort of described
by a decoupled Hamiltonian.
So that's what I wanted to present you today.
And I will show you on Wednesday how this Hamiltonian
leads to Rabi oscillations, not induced by an external field,
but induced by the vacuum.
Any questions?