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M1L8e.txt
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M1L8e.txt
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#
# File: content-mit-8422-1x-captions/M1L8e.txt
#
# Captions for 8.422x module
#
# This file has 159 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
Let me sort of wrap it up.
I've given you different angles at the g2 function.
Some are simply classical intensity distribution.
Some is simply interference.
Another aspect was only counting statistics.
My understanding and my interpretation
is they look as different as they can possibly look,
but they reflect the same physics.
Because interference of light only happens because light
consists of photons and photons or bosons.
So the classical interference and
the quantum mechanical counting of bosons lead--
and not for random reasons, but lead for profound reasons--
to the same result. And chaotic light,
which seems to be determined by just random fluctuation-- well,
the random fluctuations, if you have single-mode light,
a thermal distribution of single-mode photons
comes from random phases.
And these random phases lead to random interference,
and so we are back to interference.
Finally, let me give you my view on the measurements of g2.
When you look through the literature,
you'll find the famous [INAUDIBLE] experiment.
In one of your homework assignments,
you look at a seminal experiment where people dropped atoms out
of cold atom clouds and measured the g2 function for cold atoms.
Sometimes it's confusing when you directly compare the two
experiments, but let me try to give you
a common description, or the common denominator,
between all these experiments.
You can say that all experiments to measure
the two-particle correlation function
is about comparing the probability of finding
two particles with a probability of finding one particle.
And you restrict your measurement, your detection
of particles, to either one quantum state,
one mode of the electromagnetic field, one coherence length
or coherence volume of light.
Or if you use a semi-classical argument
for the description of particles,
you take one phase space cell.
This is one phase space cell, one mode, one coherence,
one quantum state is the different definitions of what
the quantum state is, or what a wave
packet is with Heisenberg's limited uncertainty.
You cannot define a particle in phase space to be, then,
better than the phase space volume of a quantum state,
which is h squared.
Or is it h bar?
I don't think there's a bar.
H squared.
And so this is how you can relate wave packets to quantum
states, to phase space volume.
Mathematically I could say in different basis sets,
in different basis sets, it is one quantum state.
But you can use wave packets, you
can use time-dependent description.
But what I said here-- one quantum state, one mode,
one coherence volume--
this is as well as you can define a particle or photon
to be, by fundamental reasons by Heisenberg's uncertainty
relation.
What do we mean by one coherence volume?
Pardon?
What do we mean by one coherence volume?
Well, if you have a laser beam which propagates
and it is [? Te ?] and zero, zero,
you would say the transverse coherence volume
is the size of the laser beam.
But then the longitudinal coherence volume
is given by the coherence length.
So just envision you have a laser beam here.
And the coherence volume, where all the photons are coherent,
is the area of the laser beam times the coherence length.
And whenever you find a photon in this kind of volume,
then it is in a Heisenberg uncertainty,
limited or Fourier-limited state.
If it's a wave packet in pulse laser,
it's in a time-dependent quantum state, but it is as coherent.
It is fully coherent.
So you have to figure out for your system what
is the coherence volume, what defines the fundamental phase
space cell of your system.
And now you do the following.
You are asking, what is the probability
to find two particles and what is the probability
to find one particle.
So if P is the probability to find one particle,
then you have three options--
p squared, 2p squared, or zero to find two particles.
One is the classical case of distinguishable particles.
This is the case of bosons, bosonic atoms or photons.
Therefore it includes the [INAUDIBLE] experiment,
and zero is fermions.
And the question is now, different experiments,
how do you define with spatial resolution
or temporal resolution the coherence volume?
And that can involve transverse collimation,
temporal resolution, and whatever
you can use to define the phase space
cell or the coherence length, the coherence volume of a laser
beam.
Let me give you one example, which I think illustrates it.
And this is the example of an atom cloud.
If you have an atom cloud, all the particles,
which are in a volume of a thermal de Broglie wavelengths,
are coherent.
The momentum uncertainty of particles in a thermal cloud
is the thermal momentum.
And according to Heisenberg, the position uncertainty
related with this momentum spread
is just the thermal de Broglie wavelengths.
So you can say that all the atoms in a cubic de Broglie
wavelengths are coherent.
All the atoms are in one semi-classical quantum state.
So therefore, this is sort of atoms in one single mode.
So the [INAUDIBLE] experiment with atoms,
or the measurement of the g2 function,
could be defined as follows.
You have an atom cloud and if you had an electron microscope
or some high resolution device, you grab into your cloud
and ask what is the probability for one particle, what
is the probability for two particles.
And you will find that p2 is two times p1 squared.
If your volume is too big, you lose the factor of two,
because you average over uncorrelated volumes.
Now, in your homework, you were looking
at the question, how can I really grab into a cloud
and just pick out atoms out of one phase space
cell of effective size, lambda de Broglie cubed.
And how it was done is that you take an atom cloud
and you drop it and expand it.
When the cloud expands, there is a mapping from momentum space
into position space.
Then you use some form of pinhole, which provides
transverse collimation.
You use a detection laser, which gives you temporal resolution.
And, well, this was part of your homework,
but I just wanted to give you sort of the bird's eye view
on it.
By controlling the transverse collimation
and the temporal resolution of the detection,
you create a situation that what you count
are only atoms which originated from one phase space
cell in your cloud.
So in other words, that's an experiment
where without electron microscope,
without sub-micron spatial resolution,
you can literally grab into a cloud,
capture a volume of the thermal de Broglie wavelength,
open your hand, and figure out how
is the probability for two particles related
to the probability of finding one particle.
Any questions?