M3L16a.txt

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In the semi-classic classical description
of the electromagnetic field, photons
can only be emitted, because we have
a Hamiltonian with a semi-classical electric field.
So if you don't drive the system with an electric field,
you cannot stimulate the emission of photons.
But we know this is not what happens.
Photons are emitted into empty space.
Photons are emitted into vacuum.
And for that, we needed a quantized description
of the electromagnetic field, with a field quantization.
And we have now our quantized Hamiltonian
And on Monday, I started to discuss
what is sort of the paradigmatic situation,
the paradigmatic example for how you should think
about the vacuum, and how you should think about emission
of photons into the vacuum.
And these are the vacuum Rabi oscillation, described
by the Jaynes-Cummings model.
So the situation, which I have in mind,
or which you should have in mind is, an idealized situation,
but it has been realized experimentally.
And some of those idealized experiments
were recognized with the Nobel Prize
to Serge Haroche and Dave Wineland.
So the situation is, we have an atom,
but it only talks to one mode of the electromagnetic field.
And we make sure that the atom only
talks to one mode of the electromagnetic field.
Not by eliminating all the modes, they exist-- I mean,
an atom can emit upwards and downwards,
but we sound it with a cavity, which has such a small mode
volume.
It has such a small volume, that the single photon Rabi
frequency is huge.
And therefore, the emission into this one single mode
dominates over the emission into all other modes.

So this is the condition that the single photon Rabi
frequency has to be larger than gamma.
And, of course, we also have to make sure
that the system is idealized, so the loss of photons, because
of losses in the mirror, or finite reflectivity
in the mirror, also has to be smaller.
So that means, for several Rabi periods,
we have a system which has only two parts, a two-level atom,
and one single mode of the cavity.

So that's the system we have in mind.
And we discussed the Hamiltonian.

We saw that the Hilbert space of the atom
is excited in ground state, the Hilbert space of the photons
is spanned by the Fock states.
But what happens is-- so there's an infinite number of states,
because of an infinite number of states of the photon field.
But what happens is the Hamiltonian couple only
an excited state with n photons to ground state
with n plus 1 photons.

The whole Hilbert space is segmented now
into just passive states, labeled by the index n.
So after so much work, we are back to a two-level system.
And here is our two-level Hamiltonian.
And well, a two-level system does oscillations
between the two levels, Rabi oscillations no surprise,
and this is what I want to discuss now.
But the new feature is that these are now
really two levels.
Each of them is the combined state
of the atom and the quantized radiation field.
So now, we have included in our two-level description,
the quantum state of the electromagnetic field.
So first, you should realize that this Hamiltonian
is absolutely identical to spin 1/2 in magnetic fields.

And you can recognize by comparing this Hamiltonian,
this matrix, to the matrices we discussed for spin 1/2
in the magnetic field, that this corresponds
to the situation, where the spin 1/2 had a transverse field
in the x direction, which caused the precession from spin
up to spin down.
And this x component of the field
corresponds now to the single-photon Rabi frequency
times n plus 1.
That's the off-diagonal matrix element in this matrix.
The thing which we have to discuss,
and I will focus later, is that it depends on n.
So for each pairs of state labeled
by n, the photon number, we have a different off-diagonal matrix
element.
But let's discuss first the most important and simplest case.
Let's assume we are on resonance.
And we want to assume that we're vacuum.
Then our Hamiltonian is simply this.
And when we prepare the system in an initial state, which
is an excited state with no photon in the vacuum,
then we'll have oscillations to the ground
state with one photon.
These oscillations are exactly the oscillations we saw
in the spin-1/2 system.
We can just map the solution.
I'm not really [INAUDIBLE] anything here.
So what we obtain is the famous vacuum
Rabi oscillations, where the probability to be
in the excited state oscillates with the single-photon Rabi
frequency omega one.
I think that's a little of a an ambiguity in language.
Is it the single photon Rabi frequency,
or is it the vacuum Rabi frequency?
Because there's always the question about plus
minus one photon, because we stat in the excited
state without photon, so you want to say it's
a vacuum Rabi frequency.
But then you have the ground state with one photon.
And this photon is reabsorbed, and then you
may want to call it the one-photon Rabi frequency.
So I leave it to you, but it's called vacuum Rabi oscillation.
And this Rabi frequency is usually
referred to as the one-photon Rabi frequency,
because we obtained the Rabi frequency
by calculating the electric field of a single photon.

So the Rabi oscillations, which we are observing now,
correspond to the periodic spontaneous emission
and reabsorption of the same photon.
There's only one photon which is spontaneously
emitted and reabsorbed in a completely
reversible coherent way, and the time evolution is unitary.
So it's a periodic spontaneous emission, and absorption
of the same photon.