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M5L22d.txt
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#
# File: content-mit-8-421-5x-subtitles/M5L22d.txt
#
# Captions for 8.421x module
#
# This file has 99 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
When we measure the phase and we find
that there are fluctuations, they actually
come from the quantum nature.
They come from the quantum nature of the states involved.
OK.
So let's talk about the measurement,
and just sort of let me set it up genetically.
Here's our atom.
Here is the laser beam.
And we want to create a Mach-Zehnder interferometer.
Let's do it another color for the laser beam.
Well, why don't we need to take sodium today, which
has emission in the yellow near the orange.
So the idea is the following.
We have a laser beam which is used to excite the atom.
And here, we have a switch.
And what we let through is only a certain pulse.
Let's say, if we want to have a coherent superposition
between current and excited state,
it would be a pi over 2 pulse.
Then after the atom has absorbed the pulse,
we switch off the light paths.
So then, in the second stage, the atom can emit,
and the immediate light interferes
with the local oscillator, which is the laser beam.
And we can measure the beat node on the detector.
So this is sort of the scheme how
we do a homodyne measurement.
And so, we assume we have a very short pulse which
excites the atom.
Then we switch off the laser in the upper paths,
and the light which reaches the detector for homodyne
is only the light which has been emitted by the atom maybe
a nanosecond later.
So we do a homodyne measurement of the phase
of the wave, or the wave train, emitted by the atom.
And the distribution of measurements for the phase--
I don't want to give you sort of mathematical expressions,
but it's pretty much what you can read from the drawing I've
shown you.
So for a pi over 2 pulse, well, we retrieve the phase phi,
but with fluctuations.
Let's now come to the interesting case
that we have a pi pulse.
The pi pulse prepares the atom in an excited state.
And at T equals 0, after the excitation,
there's absolutely no coherence.
The density operator for the atom
has just one in the column and row for the excited state.
There's no off-diagram matrix element.
There is no phase information.
So at T equals 0, no coherence, no phase.
So now we have excited the atom with a pi pulse,
but we have-- there is no phase information
in the atomic system.
And that would also mean that when we now
start mapping the quantum state of the atom
onto the quantum state of the light,
there won't be any specific phase for the light.
We could say, after the spontaneous emission is over,
before we do any measurement process,
we have mapped a Fock state of the atom
onto a Fock state N equals 1 of the photon field,
and there is no phase associated with a number state.
But let's be a little bit more specific here.
Let's assume we can have an ensemble of atoms,
we can repeat the measurement many times.
And let's ask the question, what happens
after the atom which was originally in the excited state
has decayed to 50%.
Well, then we have a wave function
which is a superposition of ground and excited state.
And there is a phase phi now, but this phase phi
is completely random.
So for those of you who are concerned that I call it a wave
function, you can be more specific
in the sense of Quantum Monte Carlo,
that at any Quantum Monte Carlo wave function, at any given
moment you have a wave function, but the ensemble
of your-- the ensemble, as in the ensemble of your atom,
is now an ensemble of all those wave functions
with a random phase phi.
And this is a way how you can decompose
the statistical operator of the system.
But the result is the phase is random.
If the phase is random, that means no coherence.
The statistical operator does not have an off-diagram matrix
element.
It also means that if you would ask
what is the ensemble average of the dipole moment--
the dipole moment is given by the Bloch vector,
value of all phases are equally populated in your ensemble--
the dipole moment average is to 0.
But of course, you have a d-squared value,
a value of the dipole moment, which is not 0.