M5L22b.txt

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I want to come back to this topic of spontaneous emission
because many of us have deep-rooted misconceptions
about what spontaneous emission is.
And we discussed on Wednesday that spontaneous emission
is not so spontaneous as many of us
assume because it's a unitary time evolution with an operator
with a term in the Hamiltonian.
And it is exactly this operator which
takes the wave function of the total system, the atoms
and the light to whatever it is later on.
There is no random phase.
There is no random variable in this time evolution.
Exclamation mark.
But there are certain aspects associated
with spontaneous emission and I want to address them.
But on the other hand, if you think
about spontaneous emission in the most fundamental way,
the first thing you should think about are
vacuum Rabi oscillations.
Here you see in the simplest possible system
what spontaneous emission can do for you.

So the way how we want to discuss an important aspect
of spontaneous emission, we want to go
beyond the vacuum Rabi oscillation is the following.
We start with an atom in the ground state
and the cavity is in the vacuum state.
But now we take a short pulse of a laser
and we prepare the atom.
And the laser, because the laser outputs a coherent state,
the coherent state is a well-defined phase.
And this phase appears in the superposition
between a ground and excited state
because the superposition is great with a matrix element,
which has the electric field of the laser.
But then we allow spontaneous emission to happen
and spontaneous emission to happen
means we take our operator which I just showed you,
we propagate forward in time in such a way
that we just go through half a cycle of a vacuum Rabi
oscillation, which means everything
which was in the excited state is now in the ground state.
And by just exactly propagating this system forward in time,
we obtain this state and that's sort of something
I hope very, very insightful which
we arrived at the end of the last lecture
that the quantum state of the atom
has been perfectly mapped onto the photon field.
So all of the information which was
in the atom before spontaneous emission
is now available in the photon field.
So the next thing to address is the phase phi.
What is the phase of this spontaneously emitted
photon and this is what you want to understand now.