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M1L8c.txt
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M1L8c.txt
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#
# File: content-mit-8422-1x-captions/M1L8c.txt
#
# Captions for 8.422x module
#
# This file has 118 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
The third view is classical versus quantum statistics,
and I think this really shows that we
need at least some quantumness to find correlations.
Let's assume we have n particles and we have n possible states.
And if it's a classical system, let's
assume we want to find one particle in a certain state.
We have a big box, have a small subvolume,
and say, what is the probability of finding one particle here.
So this probability to find one particle is P1,
is small n over big N. The probability
to find two particles, if we have
non-interacting classical particles, well,
is simply P1 squared.
Independent classical particles-- they
don't care what they are doing.
You grab into a subvolume, which is maybe a phase space
cell and therefore your quantum state,
your probability P of finding one.
But since each particle moves around independently,
the probability to find two particles is just P squared.
It's a little bit like if you toss
two coins, one with your left and one with your right hand.
What is the probability to find head or tail,
and you simply multiply the probabilities.
But now, following reasonings which Bose and Einstein
introduced, you want to use counting statistics, which
takes into account the indistinguishability
of particles.
So if we go from the classical distribution
to the distribution of indistinguishable particles,
then the classical probability, P1, is reduced by n factorial.
The probability to find one particle
is represented by a microconfiguration classically.
And each configuration, when you permute particles,
counts as independent.
But if you have quantum indistinguishability,
you're not counting permutations as an independent
configuration, and therefore you have a reduction
by n factorial.
If you look at the probability of finding two particles,
the reduction is-- this is just the counting statistics--
n factorial by two.
So therefore, you will find that classically, that quantum
mechanically, for bosons is two times-- the probability
to find two bosons in one quantum state
is two times the probability squared to find one boson.
So this is just counting statistics.
This can be applied to collisions in a Bose gas.
When you have inelastic collision, spin relaxation,
dipolar collisions, two-body collisions,
two-body collisions have a rate, gamma two,
which is proportional to the probability of finding
two particles at the same time at the same location.
And that brings in the g2 function.
Whereas three-body collisions, gamma three,
reflects the third autocorrelation function, g3,
which is defined in an analogous way.
So if you now compare at the same density
the two-body rate coefficient-- the same density means
we are looking at the rate of two-body collisions--
and we compare between two-body collisions
in a thermal cloud, which has a g2 factor of two
at thermal cloud of bosons.
Well, the Bose-Einstein condensate
is a constant occupation of particles in one mode
and has therefore a g2 factor of one-- exactly as the laser.
The Bose-Einstein condensate is for particles, for metawaves,
but the laser is for light.
Whereas if you look at three-body collisions,
those who do experiments with Bose-Einstein condensates
will know that usually the lifetime
of Bose-Einstein condensates is limited
by three-body collisions.
Well, I didn't derive it, but actually, yes, we
have the Gaussian statistics.
For randomness for Gaussian statistics,
the n-body correlation function has a factor
of n factorial, so therefore three-body collisions
scale with three factorial and two-body collisions with two
factorial.
So in other words, you have a thermal cloud,
you have a Bose-Einstein condensate at the same density.
But what matters for two-body collision
is not the average density.
It is the average of n squared.
And the fluctuations in the thermal cloud,
because of the g2 function, are two times enhanced
compared to Bose-Einstein condensate,
and therefore you find that at the same density
you have more loss, you have a higher rate
of two- and three-body collisions.
Some of this became better understood soon
after Bose-Einstein condensates were realized.
The Boulder group studies three-body collisions.
And for two-body collisions, myself
and the post-doc clarified the situation
that even the mean field energy, which everybody had measured
before, requires two particles to interact
with short-range interaction, is therefore proportional to g2.
And therefore, when people had determined the mean field
energy of a condensate without really knowing,
they had already determined the g2 function
of Bose-Einstein condensates.
So this is pretty much the counting statistic,
which you do in an undergraduate class
when you derive the three different statistics--
Boltzmann statistics Bose-Einstein statistics, Fermi
statistics.
And the result of that is also for Bosons,
you have the factor of two.
For fermions, of course, you get zero.
The average between bosons, two, and fermions, zero, is one,
and this is the classical statistics.
Any questions?