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M5L22o.txt
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#
# File: content-mit-8-421-5x-subtitles/M5L22o.txt
#
# Captions for 8.421x module
#
# This file has 113 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
OK.
Let us assume we have the situation we discussed earlier.
We have a quantum beat where we have a beat frequency omega 0.
Now just think about the [? searchlight, ?] the atoms
which oscillate with [? a lot more ?] frequency,
and you have some cosine omega [? a lot more ?] t vector
in the intensity of the light you observe.
But now, because the atom in the excited state is decaying,
everything will decay with the natural line widths.
And this is sort of what we observe.
And the question is if you observe that in real time, can
we then retrieve spectral information from it which
is more accurate than gamma.
So all I want to do is I want to discuss the Fourier transform
of this function S of t.
Let me use dimensionless variables.
We measure frequencies in unit of gamma.
We use a Lorentzian, which is just 1 plus x squared.
And-- OK.
We just said we want to start the measurement at time t0.
So the question is, if you start later and later and later,
do we get higher accuracy because we're
talking to the survivors?
So let's perform the Fourier transform.
And let's use complex notation, e to the i omega t.
And I will measure times in units of the inverse line
widths.
We perform the Fourier transform.
And by doing e to the i omega t, I actually
perform the Fourier transform for the cosine and for the sine
by using the real part and the imaginary part
of the complex number.
So we will actually be able to look
at the real and the imaginary part.
And the Fourier transform has a real and imaginary part.
So let me call the real part F of x, and the imaginary part
G of x.
In both cases, will we find, because of the-- I mean,
you can do the math.
It's a straightforward integral.
In both cases, will we find that, if you
do our measurement, well, the longer we wait,
the more signal we lose.
I mean, this is common to all delayed measurements.
You're really now talking to an exponentially smaller
and smaller signal.
Also, because of the exponential decay,
we get an envelope which is a Lorentzian.
But then-- and this is the interesting part--
we have cosine xt minus [? x ?] sine xt.
So now we have [? factors ?] which
depend on capital T. And T is larger the longer we wait.
And it is actually those parts with the sine and cosine which
determine whether we can get resolution
below the natural line widths.
OK.
What is important is-- and this should
be sort of an eye-opener for you--
if you simply measure intensity, if you look at the power
spectrum, you take the real part plus the imaginary part
and just look at the absolute value, then because of cosine
squared plus sine squared equals 1,
all the cosine and sine parts, the last part of the expression
above, cancels out.
And you find that you have an exponential loss of signal.
But your spectral distribution is always a Lorentzian.
So you always have a Lorentzian line shape completely
independent of the delay time capital T.
And this is what some of you maybe thought.
If I start the measurement later and all
I can do is look at the power of the immediate light,
I have no advantage.
I cannot go [? sub-natural. ?]
However, if you look at the function
F of x for large values of x, you find oscillations.
So if you look at the sine or cosine Fourier
transform, the real or imaginary part separately,
you find oscillations.
And those oscillations, similar to Ramsey fringes,
have a central peak.
The central peak is narrow.
And the widths is now given by not the inverse
of the natural lifetime, but the inverse of the delay time
you wait for your measurement.
So the fact is now we had a signal, S of t,
which we assumed was a quantum beat with a well-defined phase.
And then it was exponentially decaying.
If we would now perform a Fourier transform where
we do a Fourier transform with cosine omega t plus phi,
we can get a narrow signal.
But if you have no idea of what phi is,
or if in repetitions of the experiment phi would be random,
then-- I mean, this is sort of what the math does,
is if phi is random, it is the same
as if you simply measure the power spectrum.
Because we cannot distinguish between the cosine and sine
Fourier transform.
So if the phase phi is random, that
means you only measure what was F plus IG before.
In other words, the situation is actually
in the end very simple.
If you have quantum beats which start
with a well-defined phase, and you know the phase was,
let's say, 0 here, and now you have the decaying function,
and now you look at the quantum beat over there, well,
in a way you had sort of n beats between T equals
0 and your measurement.
And then your resolution goes with 1 over n.
But you have to know what the phase was at T equals 0.
And then, by looking at delayed detection,
you can do [? sub-- ?] spectroscopy