M1L3g2.txt

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I want to now-- I want to in a minute draw you
a table, what are the fluctuations in the photon
number for the states we have encountered,
the number state, the Fock state,
the coherent state, and the Thurman's state.
And we want to characterize those quantum states of light
by the g2 function.
And actually, I can drop the time argument,
or set it to zero.
But then we first make a reference
that g2 can now be smaller than 1.
So g2 for a single mode light is nothing else than a function.
When you know what is an average, an n-square average,
you know your g2 function.
There is another quantity which we often
use to characterize the fluctuations in the photon
number, and this is called the Fano factor.
The Fano factor wants to compare the fluctuations n-square
average, minus n-average squared.
These are the fluctuations.

The classical fluctuation, well not
say classical, classical in the simplest case
are Poissonian fluctuations.
So maybe we want to normalize the fluctuations
by Poissonian fluctuation.
So for Poissonian statistics, what I just wrote down
would be one.

And well, if we now subtract one,
we have the situation that the Fano factor, which is positive,
is super Poissonian, more fluctuation than Poissonian,
and the negative Fano factor is sub-Poissonian.
So with those definitions we can now
compare the different states of light we have introduced.
We started out with black body radiation, thermal radiation.
We defined coherent states, and the harmonic oscillator
description naturally gave us harmonic oscillator
eigenstates, the number states, or Fock states.
So there are three ways to characterize it.
They are all useful.
One is we can look at n-squared.
We can calculate the Fano factor,
or we can calculate g2 of zero.

For the coherent state, remember that the coherent state
is as close as we can come quantum-mechanically
to the ideal of a pure electromagnetic wave.
It has a Fano factor of zero.
This means it's Poissonian.
The g2 function is simply one, which
is the lowest classical limit.
So those two tell you that a coherent state is sometimes
what you think would come out of a laser,
is an ideal electromagnetic wave,
which has no temporal fluctuations in the intensity.
Therefore, g2 is one, and the photon number
is Poissonian distributed.
And that means that n-squared is an average times 1
plus an average.
We actually discussed it earlier.

The thermal state is quite different.
If you use-- kind of pluck together
the results we have obtained, it has a Fano factor of n-bar.

So this is super Poissonian.
If the occupation number n is large,
you have fluctuations which are much,
much larger than Poissonian fluctuation.
The g2 function is 2.
It's sometimes called thermal light.
Chaotic light has a g2 function of 2.
Laser light or coherent state has a g2 function of one.
And n-square average is n-bar 1 plus 2 n-bar.

OK but now finally, maybe the most interesting state
from the perspective of non-classical light of quantum
light, is the photon number state.
Well, for photon number state, the number of photons
is an eigenvalue.
Therefore, n-square average is an average squared.
The Fano factor is minus 1, sub-Poissonian distribution.
And the g2 function, which classically cannot go below 1,
is now n minus 1 over n.
It is smaller than 1.
And you see it immediately that is the biggest violation for g2
is zero for the case of a single photon state.