M5L23h.txt

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I want to give you a quantitative touch
without doing too much math.
So let me just write down Schrodinger's equation.
We have the three amplitudes, the three states involved,
E, G, F. Schrodinger's equation gives us
the time derivative, c.f.

Since we assumed-- in the [INAUDIBLE] state
picture, we can assume everything is degenerate.
But now we have couplings.
And the only couplings are-- and this
is what I was emphasizing-- between ground and excited
state with a Rabi frequency omega 1 and between the state f
and the excited state with a Rabi frequency omega 2.
So the important part is-- I can write it down,
but the third equation is the one
I want to focus on, that you can only build up population
in the final state by transferring with a Rabi
frequency omega 2 population from the excited state.
So let me just assume that Rabi frequency omega 1 equals
omega 2 equals omega Rabi.
Let's have a symmetric situation so I can now completely focus
how can you get from the excited state to the final state,
but how you get from the ground state,
from the state g to the excited state, is fairly symmetric
here.
OK.
So what we want is we want that at the end of the transfer
time, t transfer, the final state
amplitude is on the order of 1.
This would be 100% transfer.
So therefore, just looking at the differential equation
above, that means that this here has to be 1 divided
by the transfer time.
So therefore, we find that for this adiabatic transfer,
the excited state amplitude has to be [INAUDIBLE] factors
on the order of two now has to be
1 over the Rabi frequency times the transfer time.
So the probability to be in the excited state
is the amplitude squared.

And if you're asking, which I think is a practical question
and I think this-- I suggest to use this as a figure of merit
when we discuss how well, how perfect have we transferred--
how perfectly have we transferred population.
We can now ask what is the probability
of spontaneous emission.

P spontaneous is the probability to be
in the excited state times gamma integrated over time.

So what we obtain for that is, yes, it's
proportionate to gamma.
But we have the excited state amplitude squared.
We've multiplied with the transfer time.
So therefore, what we have in the denominator is
[INAUDIBLE] power or the Rabi frequency
squared times the transfer time.
So therefore, this probability to go
to a spontaneous emission, which is also the integrated
probability of having-- the integrated
probability of having populated the excited state,
it goes to 0 in the case that you have infinite laser power
or that you are infinitely patient.
So one question is how long can the transfer time be.
And this is actually setting a limit to it.
We're talking about a coherent transfer.

So the transfer time has to be smaller than the coherence
time of your system.
For instance, we do that all the time
in my lab if you go from one sub-level
to another sub-level and we have magnetic field noise.
If the magnetic field noise is destroying the phase
relationship between the two hyperfine states,
then the coherent transfer gets interrupted.
And then eventually we are no longer
talking about the amplitude, we are talking about population
in the excited state.