M1L3d2.txt

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So I want to sort of give you, in a two-dimensional diagram,
a visual representation of thermal states,
coherent states, number states.
And this visual representation will
make it immediately obvious what the electric field is.
So this visual representation will represent what
are called quasi probabilities.
So I have to define for you now those quasi probabilities.
So I denote the quasi probability by Q.
And I want to know what is the quasi probability
Q for any statistical operator which describes a light.
It could be the operator for thermal light.
It could be statistical operator also includes, of course,
pure state.
And the pure state could be a coherent state.
So this is a way to define our quasi probabilities
for any statistical operator which is describing the light.
And the quasi probabilities-- well, the probability for what?
Well, the probabilities for alpha.
And alpha is related to the coherent state.
So in other words, we say a statistical operator
has a quasi probability with a coherent state alpha by simply
calculating this diagonal matrix element
of the statistical operator with alpha.
That's an abstract definition, but it's exact.
What we should now do is, I want to show you some examples.
And then you will actually see that it's actually
a wonderful definition.
So my first example is, let's look at the quasi probability
for the vacuum state-- so if the statistical operator is simply
the ground state of the harmonic oscillator--
no photon, no nothing, just empty vacuum.
So the quasi probability for-- the statistical operator
is now a pure state, so I can say
what is the quasi probability of the vacuum state
to be in state alpha.
Following the definition above, it's
nothing else than the matrix element, the overlap,
between the vacuum and the coherent state alpha.

I've given you above the representation
of the coherent state alpha in terms of number states.
And all you have to do is look now
upstairs, what is the amplitude that this representation
includes a vacuum state.
So it's just read off the amplitude c0 from the state
above.
And this was simply e to the minus alpha squared.

Those quasi probabilities can be plotted
in a two-dimensional plane, because it's
a function of alpha.
So if I use the real part of alpha, the imaginary part
of alpha, now I can plot the quasi probability Q.
And since it's hard for me to do a 3D plot on the tablet,
I just sort of shade the region.
Black means the amplitude is large.
And now it's sort of here, it means the amplitude falls off.
So what I'm plotting here is a Gaussian.
So here we have a Gaussian.
Any question?
So for a pure state, we are simply
asking, what is the amplitude of the pure state to overlap,
to be in a coherent state?
And what we are plotting is nothing
else than the probability that this state is a coherent state
with value alpha.
Further question?
The next example-- and we have already
prepared for that-- is the thermal state.
Example Number 2 is the thermal state.
So we want to know what is the quasi probability
for the thermal state as a function of alpha.
The statistical operator for the thermal states--
we had derived it above-- was the probability
to find n photons.
And in the number representation,
the statistical operator is diagonal.
And then the quasi probabilities are nothing else,
following the definition above, than those probabilities
times the matrix element-- the overlap between alpha and n.
And if you look up what we had derived for the probability Pn,
you find a very simple result. What we obtain
is, again, a Gaussian.
It is also centered at the origin.
So therefore, since I didn't label my axes,
it looks exactly as the vacuum state.
It's just that the Gaussian is much, much broader.

And I will come to that.
But an intuitive reading of this Gaussian
is, this Gaussian has a peak at the origin.
At the origin, this is where alpha equals 0.
And this is directly related to the question I asked you
before, that the photon number probability is
peaked at n equals 0.
But I hope this will become even clearer when
we move on to the third example, which is now Q
of alpha of the coherent state.

Maybe just before I even discuss the quasi probability
of a coherent state, I have to address one thing up front.
What would you expect the quasi probability of a coherent state
to be?
[INAUDIBLE]

Pretty good, but I said before the quasi probability is-- let
me just show you how it was defined.
Here was my definition of the quasi probability.
And you can see the quasi probability
is the diagonal matrix element of the statistical operator
with a coherent state.
But if our statistical operator is a coherent state, what would
you expect this probability to be, just naively?
One.
One or delta function.
I mean, we ask if we expand this statistical operator
in coherent states, what do we get?
But if this operator is a coherent state--
let's say it's a coherent state beta-- you would expect then
we only get something on vanishing if alpha equals beta.
So you would now even expect that what you get
is a delta function.
The coherent state beta has a quasi probability
Q of alpha, which is peaked at beta,
but it's peaked as a delta function.
Now, I'm calculating it for you right now,
and the result is, this is not the case.
The reason is-- and I will show you
that [INAUDIBLE] is-- that the coherent states are not
your ordinary basis function.
The coherent state forms a basis which is over complete.
So we have sort of more coherent states than necessary.
There is some redundancy.
And therefore, the coherent state alpha
and the coherent state beta-- if alpha
and beta is only-- if there's only
a small difference between alpha and beta,
they have overlap-- coherent states
which differ by only a little bit in the eigenvalue.
They are not orthogonal.
So this is a complication.
It's one of the complications we have
to deal with when we want to describe coherent states
and when we want to describe laser light.
But this is one of the properties.
So therefore, what I'm now discussing
with you is what are the quasi probabilities
of a coherent state.
It will not be a delta function.
But we should simply follow the definition,
and everything will fall into place.
So our statistical operator for a pure coherent state
is, of course-- this is a statistical operator
with pure state beta.
And the quasi probability for the coherent state beta
to be in alpha is this matrix element squared.

And you can calculate that, for instance,
by just putting in now for alpha and beta the expansion of alpha
and beta in number states.
So you just have to solve the integral and calculate it.
And what you find is not a delta function,
but a decaying exponential.
So you can say as long as alpha and beta do not
differ by more than 1, there is substantial overlap.
So in other words, what we get for this quasi probability is
something which is centered at beta,
but it is, again, a Gaussian.
So the important message here is, coherent states are not
orthogonal to each other.

They form an over complete basis.

This is a side remark.
It may come as a surprise to you,
because you have been trained too
much in basis states of emission operators,
of energy eigenstates.
And those energy eigenstates form a complete basis.
The coherent states are eigenfunctions
of a very strange operator-- a non-Hermitian [? inoculation ?]
operator.
So therefore, the few things you took for granted
do not apply here.
So we have our nice diagram for quasi probabilities.
So now, that was the real part of alpha.
That was the imaginary part of alpha.
So if you take that as the complex plane--
and I'm drawing now as a phasor the complex number beta, which
is the eigenvalue of the coherent state beta--
then the quasi probability is centered near beta.
And it's a decaying Gaussian.

I'm not showing it to you here explicitly,
but it is simply the vacuum state we discussed before,
displaced by beta.

My fourth example, just to show you-- so far
I've shown you that everything is Gaussian.
Thermal light is a broad Gaussian.
Coherent state is a narrow Gaussian.
The vacuum state is a narrow Gaussian.
Let me just show you something which is non-Gaussian,
which is the number state.
If you put into the definition of the quasi probability
the number state, and you calculate the quasi probability
as a function of alpha, then you find that what you get
is actually not a Gaussian.
What you get is a ring with a certain width.
So this is the representation of a number state.
And the radius of this ring is proportional to the square root
of the number of photons in the number state.
We'll come back to photon states in much more detail very soon,
but I just wanted to show you that at this point.
So quasi probabilities are not always Gaussian.
They really depict something.
They show us important differences about the quantum
states of light.
Any questions?
Yes.
[INAUDIBLE] a delta function in the radius?
Like you went a little past to 0 [INAUDIBLE]?
Can you hold the question for three minutes?
OK, the answer is, it's not.
But in three minutes, I want to tell you
that there are three possible definitions
of quasi probabilities-- the Q function,
the W function and the P function.
In one of the functions, it's a delta function.
In the Q function, it's not.
But we'll come to that.

But let's, for now, stick to the Q function.
The next thing we want to put in is the time dependence.
I want to show you that when we have quantum states of light--
and right now, we've just shown the quasi probability
distribution at a snapshot at t equals 0.
What happens as a function of time?
What I want to for you is-- and this
makes those quasi probabilities also nice and intuitive--
that as a function of time, the quasi probability distribution
simply rotates with an angle of frequency of omega.
So let me show that to you.
We know that we have a coherent state.
And what we want to understand is
what happens when we act on the coherent state with the time
propagation operator, which is the Hamiltonian
in the exponent.
Well, in order to evaluate the left-hand side
we can simply take the expansion of coherent states
in the number basis.

Because we know that the time evolution of a number state
is simply the energy of the number state, which
is given by the energy of the number state, which is n times
h by omega.
So now, if you look for a second at this expression,
you realize that we can absorb the time evolution
if we redefine alpha to become alpha times e to the i omega t.
So in other words, the time evolution
of the coherent state alpha preserves the character
as a coherent state.
It just means we get a new coherent state
whose eigenvalue is now alpha times e to the i omega t.
So that means the following-- if we
want to look at the time evolution in terms
of quasi probabilities, we have our diagram
with the real part of alpha, with the imaginary part
of alpha.
And let's assume we had a coherent state, which
happened-- this is sort of now my circle.
It's, you can say, a high contrast
representation of a Gaussian.
So this is a quasi probability of the initial state alpha.
But as time propagates, it just gets multiplied with e
to the i omega t.
That means the time evolution is just displacing
the state over there.
And as time goes by, the coherent state
is just moving in a circle.
And after one full period of omega,
we are back where we started.
Let me give you now an intuitive picture
what the electric field is.
I will be a little bit more exact in two or three minutes.
So we have a diagram of the real part of alpha
and the imaginary part of alpha.
For an harmonic oscillator, I can also label that as x and p.
You know, in an harmonic oscillator--
you can think about it classically-- if you have
some initial distribution of a mechanical oscillator in x
and p, what the system actually does is,
it simply rotates on the circle in phase space.
The classical phase space is x and p.
And I haven't really told you that.
I didn't want to get lost in complex definitions.
I defined for you the quasi probability,
but the quasi probability is a generalization
of the classical phase space function
as a function of p and x.
So take my word for a second, and allow
me to identify the real axis with x and the imaginary axis
with p.

Then p is the electric field.
So now what you should visually take from those pictures
is that if you want to know what the electric field is,
you just sort of project this fuzzy
ball on the imaginary axis.
So in other words, when I ask you
what is now the electric field as a function of time, at t
equals 0, I project this onto the p-axis.
And what I get is something which is centered at 0.

But it has also a certain fuzziness.
The fuzziness is given by the size of the disc
or by the width of the Gaussian.
And if I now use this picture as a function of time,
as this fuzzy ball rotates around,
the electric field goes up and down in one cycle.

And the fuzziness moves with it.

So the fuzziness here for the electric field is related,
and that's what we want to discuss next week--
is related to shot noise.
I also want to tell you that the coherent state is,
to some extent, the best possible way
to define an electric field.
It's a minimum uncertainty state.
I want to show you that the coherent state is
at the minimum of Heisenberg's uncertainty relation.
So I think if you take this picture,
you will immediately realize that you
draw the quasi probability.
You project on the vertical axis.
And you get the electric field.
That tells you immediately if you have a thermal state, which
is a Gaussian centered here.
You project it.
You get an electric field, 0.
It has an enormous fuzziness, but it's 0.
And this picture-- that everything rotates with angular
frequency omega-- also tells you, as expected,
that in a thermal state, it's already a circularly symmetric
distribution.
When it rotates, nothing changes.
[INAUDIBLE] operator is [INAUDIBLE]
couldn't you find eigenstates of that operator
itself, and then get rid of this fuzziness?

What I've told you here, with the projection
of the electric field, this is not
100% rigorous for the quasi probabilities
Q. Due to the non-commutativity of operators,
when we generalize phase space function into quantum
mechanics, we can order operators an a symmetric way,
in a normal way, in an anti-normal way.
And therefore, there are three different quasi probabilities.
The Q probability is the easiest to define.
It comes in very handy.
But what I showed you with the projection
does not rigorously apply to the Q distribution.
It applies to the P or W distribution.
But the differences, unless you really go into subtleties,
are minor.
So intuitively, what I did is correct.
And to answer your question, an eigenstate
of the electric field-- I have a little bit problems with that,
because if you take this projection, what you need
is you need something which is very, very narrow.
But since this area is an Heisenberg uncertainty limited
area, it means delta x delta p equals h bar over 2,
the only way you can create this electric field
is you can completely squeeze it.
So to the best of my knowledge, an electric field eigenstate
would be a completely strongly squeezed state.
We talk about that next week.
But then you realize, if you've completely squeezed your state,
after a quarter period, this infinitely squeezed ellipse
is now standing upside down.
And after a quarter period, your electric field
is completely uncertain.
So I think you can define an operator-- you
can find a state-- which has a sharp value
of the electric field at one moment of time.
But then it rotates around.
And what was a highly certain electric field
becomes highly uncertain later.