M5L23c.txt

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OK.
So we've talked about the dark state as a superposition
of state g and f.
But if you have a two-dimensional Hilbert
space given by two ground states, g and f,
we can now find a new basis.
We have defined the dark state, but now we
can get the orthogonal state, which is the bright state.
It has a plus sign here, and it has the omega 1 and omega 2,
the Rabi frequencies reversed.
So this state is now orthogonal.

And you can now visualize the preparation of the dark state
as follows.
We have the dark state.
We have the bright state.
The bright state, as you can immediately verify,
is strongly coupled by the laser field to the excited state.
So we have an excited state.
The laser fields-- well, it's two laser fields--
strongly couple the bright state to the excited state.
And then spontaneous emission-- two photon lights scattering--
can take you back to either of the two states.
But now you see that the concept of optical pumping,
which I started out with a trivial example,
is now applying in the new basis to the bright
and to the dark state.
And what happens is you would by optical pumping
populate the dark state and completely pump out
the bright state.

Just as a side remark, there are lots of subtleties.
Most of them you will probably see by yourself
by just inspecting the results.
But if you may ask yourself what happens when the Raman
resonance is not met, when the difference between the two
lasers is not exactly the difference between the two,
well then what you have is you have a superposition
of ground and excited state.
And so you have ground state and g-- the two ground
states g and f.
But the two laser fields have a phase
which evolves different from the phase between g and f.
And suddenly, what you have is relative-- I mean,
everything is relative to the laser field-- so to speak,
this plus sign turns into a minus sign.
So therefore, if you have pumped into the dark state
but now the the frequency difference of the two lasers
is slightly different from the frequency difference between g
and f, this means in essence that, as time goes by,
with a frequency-- with the detuning away
from the two-photon resonance, the dark state will now
precess into the bright state.
And it is only for the Raman resonance,
when it is exactly met, that you have a long-lived dark state,
a dark state which is the true eigenstate of this Hamiltonian.
Yes?
Would the precession also change on [INAUDIBLE] 2?

No, we have defined omega 1 and omega 2
as the Rabi frequencies of the two lasers.
Omega 1 is the Rabi frequency of the laser which talks to I
think the state g, and omega 2 is
the Rabi frequency of the laser that talks to the state f.
So this doesn't change.
It's just-- I'm just sort of telling you
how you may be able to think about it,
that if you put in all the temporal phase
factors on resonance, at least in some [INAUDIBLE] picture,
all the phase factors are zero, because the e to the i omega
t of the atomic wave function is compensated by the light atom
coupling.
And therefore, phase factors just disappear
and this state is stationary.
But if you write down what happens as a function of time
to this state and you couple it as a function of time
to the laser field, you will find out
that there is an evolving relative phase
between the light field and the atomic state.
And the essence of that is that with this, there
will be a beat node at which the dark state becomes bright
and the bright state becomes dark.