M5L24b.txt

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I showed you one possible realization.
And this is about hydrogen in a DC electric field.
If we mix the 2s and 2p states with an electric field,
we have this three-level structure.
And for laser detuned right in the middle between the two
states, the two amplitudes for excitation cancel exactly.
And therefore, you have a zero absorption feature.

However, if you put now a little bit of population into the,
let's say, the upper state.
This upper state in the wings of its profile
has still gain for stimulated emission.
And what we get is lasing without inversion.
Now you would say, OK, but that's the hydrogen atom.
Which atom is really degenerate between two levels.
And you can split the degeneracy with a static electric field.
Well, I mean, you're already an expert
at this point in the course.
If the 2s and 2p state are widely separated, well,
you add a photon.
And the photon, which is in resonance
with the 2s and 2p state creates, in the [INAUDIBLE]
atom picture, degeneracy.
Because the 2s state-- maybe I should have shown the p state
higher-- the 2s state with one more photon
and the 2p state with one photon less have the same energy.
And then you create exactly this situation.
So therefore, the way how you can
realize that in atoms, other than hydrogen,
is use an AC electric field to mix S and P states.
And I'll show you in five minutes a little bit more
in detail what I mean by that.
There is a trivial realization, which
I want to mention for lasing without inversion.
And this would be if you have a three-level system
with an excited state, two levels G and F in the ground
state, you may have an inversion for the E to G transition.
And therefore, you can get lasing
because the population in this state F is not coupled.
I mean, this is pretty trivial.
But the more subtle part, of course,
is that we can realize it using a driven system using a control
laser by creating the same situation with the bright
and the dark state.
And population in the dark state is sort of
hidden from the light and does not absorb light.
Let me just indicate that so in both those states,
we would have no absorption dark state.
Those examples may raise the question,
whether whenever you have lasing without inversion if you
can find a basis set where you have sort of inversion
again between the two levels which are relevant.
And the extra population is just hidden.
I want to make two comments about it.
This question is sometimes discussed
in the literature, sometimes in a semi-controversial way.
There are two comments about it.
One is, when you start dressing up the laser,
dressing up your system with laser beams,
you have strong control lasers.
We have two lasers here, omega one, omega two.
One is often a strong control laser.
And the other one is the weak laser
where we want to have lasing.
You have actually a time-dependent system driven
by time-dependent fields.
And once you have a time-dependent system,
it's no longer clear what the eigenstates are,
what the population are, and what are the coherences.
So you have different-- there is no longer a unique way
to distinguish what other eigenstates.
Because every state is, so to speak,
time dependent almost by definition.
On the other hand, I think the example I gave you
with atomic hydrogen, where you just
do a little bit of mixing with an electric field,
is an example where you genuinely
have less population in the excited state
than you have in the ground state.
And at least the equation tells you, even without inversion you
have a net gain.
So my own understanding of that situation
is that in many situations, you can actually
reduce it to a simple picture where you have simply hidden
population in a dark state.
But without any sort of unnatural definitions,
you may not find that in some other systems.