M5L22v.txt

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OK.
We describe this situation with a dipole Hamiltonian.

And we make the rotating wave approximation.
So the Hamiltonian has three lines, three parts.
One is we describe each of the laser fields as a single mode.
I call the frequency now omega a and omega c,
just to connect it with the operator c dagger c.
For the atom, we use the matrix-- sort of two
by two matrices.
So this is the matrix if the atom is in the ground state.
Coherences are described by that.
And of course, without any interaction with a laser field,
the atomic Hamiltonian is-- we have
atoms are in the ground state.
The state f has 0 energy.
We use that as the origin of the energy.
The state g has an eigenenergy of omega gf.
And the excited state e has an eigenenergy of omega ef.
But now the important part is that we
want to have the coupling.
And actually, I realized I was not saying it correctly.
Omega 1 and omega 2, these are the Rabi frequencies of the two
fields, and the two fields are at frequency omega a and omega
c.
So now we have the coupling between the excited
state and the ground state via photons a and a dagger.
And the coupling happens at the Rabi frequency omega 1.
And then we have the second laser field, which
is at Rabi frequency omega 2.

And we have the atomic raising and lowering operator,
and we have the photons c and c dagger.
OK.
That's a nice Hamiltonian.
It has three lines.
So the important part here, which we have explicitly
assumed, is that each of the lasers-- a and a dagger, c
and c dagger-- are only driving one transition.
So one field is responsible for connecting the state f
to the excited state, the other field
is responsible for connecting the state g to the excited
state.
In practice, this can be accomplished
by you have maybe polarization.
This is a plus 1 state, this is a minus 1 state,
and the excited state is m equals 0.
Then one laser beam is sigma plus,
the other one is sigma minus, so it can be polarized,
and the two laser beams can only talk
to one of the current states.
Or, you can have a situation that you have a huge energy
separation-- let's say you have a large hyperfine splitting--
and the two lasers are separated by frequency.

So I think I've set the stage, but I think I should stop here.
And on Wednesday, I'll show you in the first few minutes
of the class that this Hamiltonian has
a simple solution, which is a dark state, which
is a superposition state of g and f.
Any questions?