M1L3d.txt

#
# File:   content-mit-8422-1x-captions/M1L3d.txt
#
# Captions for 8.422x module
#
# This file has 249 caption lines.
#
# Do not add or delete any lines.  If there is text missing at the end, please add it to the last line.
#
#----------------------------------------

You can really appreciate the properties of coherent states.
Coherent states were introduced by Roy Glauber in 1963.
It's actually something you should wonder about.
Coherent states seems so natural.
I mean, we teach it in all courses.
And when we think about harmonic oscillator,
we often think in terms of coherent states.
But they were not immediately used
and invented when quantum mechanics
was invented in the '30s.
It really took somebody like Roy Glauber
to invent coherent states.
And that happened in the '60s.

Maybe the reason why people didn't immediately
jump at coherent states is because coherent states are not
eigenstates of Hamiltonian.
The definition of a coherent state
is actually an eigenstate, not of an Hamiltonian.
It's an eigenstate of the annihilation operator.
And of course the annihilation operator is non-Hermitian
and so it's a strange operator to look for eigenstates.
But this is what defines the coherent state.
Namely that A, acting on a coherent state alpha
equals alpha times alpha.

I'm using here the standard notation
that alpha is both the labor for the coherent state, written
down here.
And is also the eigenvalue of the coherent state,
when acted upon with the annihilation operator.
We normalize those states.
And since we are talking about eigenvalues
of a non-Hermitian operator, alpha
can be any complex number.

OK so now we have introduced a second set of basis functions
for light.
Remember, we had the basis functions N, the eigenfunctions
of the harmonic oscillator.
And now we have defined a new set of eigenfunctions,
that coherent state alphas.
Of course, the next obvious question
is how are the two representations connected?
So what we want to do now is we want
to figure out how do we write a coherent state alpha
in the N-basis in the basis of eigenfunctions
of the harmonic oscillator.
So we want to write-- this is just
going to expand the coherent state into eigenfunctions N.
So the coefficients we want to calculate
are those matrix elements, which I define as Cn.
So what we have to use is in order
to find this representation we have
to use the fact that alpha is an eigenstate of the annihilation
operator.
So we know that this is alpha times alpha.
But we can also let the annihilation operator A act
on the state N in the sum.
And the annihilation operator, of course,
does what it is supposed to do.
It lowers the Fortran number by 1.
And we get the matrix elements square-root N.
And then by just changing the summation index by 1,
we can write it is as N times Cn plus 1 times the square-root
of N plus 1.
So this is when the annihilation operator acts on the sum.
And now I continue with a lower line,
which is alpha Cn times N. So what I've found now
is I have found two ways to hide the same state.
Namely A acting on alpha.
And therefore those coefficients have to be the same.

And what I just indicated in blue
is a recursion relation, which expresses Cn plus 1 by Cn.

So every time you both from Cn to Cn plus 1,
you multiply with the square-root of N plus 1.
So if you stare at it for a moment,
you find that Cn is related to C0.

By alpha N over N factorial.
So of course you get only the relation
between the coefficients.
But if you throw in that you want to normalize the state,
this determines C0.
And then we get as a final result
the normalized representation of the coherent state
in terms of numbers states.

Excuse me, I think a square-root is missing here.
Because it appears here times N.
OK so what we have found here is now
how coherent states are written down in number-state basis.

So now we know coherent states.
And we can ask a number of questions.
So what is this?
So what did we define, and what have we now derived?
Let's look at some of the properties.

The first one is that if you look at this expression,
this is the amplitude that we have
N photons in a coherent state.
The square of the amplitude is the probability.
So the probability to find N photons
is the square of the amplitude above.
So this is that.
And this is now Poissonian distribution.
So it's immediately clear that those coherent states
have much, much smaller fluctuations in the photon
number than the thermal state we discussed before.
Because this is Poissonian whereas the thermal stated
a variance that was was much, much larger than Poissonian.

OK, that's number one.
Number two is we would like to know what is the photon number.
So we want to know what is the expectation value for N. That's
actually very easy to find.
You could, of course, get the average number of Fortran's N
by evaluating the expansion of the coherent state in terms
of number states.
But sometimes you should also try to maybe directly evaluate
the coherent state.
And the photo number operator is A dagger A.
And now you see, hey, we are lucky because A acting on alpha
is just getting alpha.
But A dagger acting on alpha on the left-hand side
gives us alpha star, the complex conjugate.
So what we obtain here is simply the absolute value
of alpha square.
And this is our expression for the average photo number.

I have already mentioned that the variance must be --
the variance must be N because it's a Poissonian distribution.
But since it is so nice to have creation annihilation operators
acting on coherent states, let's just calculate --
let's verify that you know the distribution is Poissonian
but let's verify directly that the variance of the photo
number in the coherent state is N is Poissonian.
If you need the variance, we need an average,
and we need N square average.
Let's just calculate N-square average.

So we need now the operator for N square, which
is A dagger A, A dagger A. We know already if those two
operators act on the [INAUDIBLE] that this just gives us
alpha star alpha.

And then we are left with A A dagger.

Now A dagger acts on alpha.
We don't know what A dagger is acting on alpha.
We know that alpha is an eigenstate of A,
but not of A dagger.
So therefore, we want to change the order of dagger
use the commutator.

And then of course A dagger can act here, can act here,
and we just get another effect of alpha.
So therefore, in a very straightforward way,
we find now that N-square is alpha-square times alpha-square
plus 1.

And using the result for the new photo number,
this is nothing else than N bar 1 plus N bar.

So if you now put it together, we want to know the variance.
The variance is N square average,
which we've just calculated, minus N average squared.
And, hooray, find N bar and we verify again
that the fluctuations are Poissonian.

What else can we ask?
We've asked about the statistics of fluctuation.
We've asked about the probability to find photons.
Well, what else do we want to know about light?

Electric field.

Oh, that's something I forgot to ask.
I will come back to that later, but it's just sort of
too connected.
What is the average electric field for thermal light?

Zero.
What else can it be, because any value of the electric field,
whether the electric field is positive or negative,
requires that it -- I say it loosely --
that the photon has a phase.
The phase will define whether it's positive or negative.
But all different phases of the photon field are different --
what I mean now is coherent states which is different
phases.
They are all degenerate.
So in the thermal ensemble, there is no thermal ensemble.
The states are only populated due to their energy.
They are populated by Boltzmann factors.
And therefore there is no distinction what
the phase of the state are.
So therefore we haven't defined yet what the phase of a photon
is, therefore I'm a little bit guarded with my language.
But in other words, whatever the state with a phase
is in the thermal ensemble, you have all phases simultaneously.
And therefore you never have an expectation value of E.
But of course we had an average number of photons,
and that means we have an expectation value of E square.
Because the expectation value of E
square is the expectation value of the energy.
The state has energy, but it doesn't have an expectation
value for the electric field.
OK so maybe with that motivation we want to figure out now
does this coherent state which we have constructed,
does that have now an expectation value
for the electric field?
So there is one thing we of course
know about this electric field.
Whenever we have an electric field
it needs an amplitude and a phase.

And what I just mentioned to you is that in thermal ensemble,
we were just looking at the partition function and such.
We didn't even mention a phase.
There was not any complex number in any of the equations
we wrote down about thermal light.
No complex number, no phase, means no electric field.
But this is different now for the coherent state.
Because I mentioned that alpha in general can be complex.
The annihilation operator can have complex eigenvalues.
So therefore in the coherent state,
because it is characterized by a complex number,
we have now the possibility to include the phase.

And what I want to show you is, in the next few minutes,
is that yes indeed, the coherent state has an expectation
value of the electric field.

Instead of simply using the equation for the electric field
operator, remember electric field operator
was A minus A dagger.

I want to introduce something else, which will turn out
to be very, very useful.
And one of the application of this concept
is to show you what the electric field is.
This is sort of a visual representation.
A visual representation of the states