M1L3f.txt

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Just to give you the bigger picture, what
we want to achieve with those quasi probabilities,
we want to do what phase space densities in classical physics.
We have a coordinate which is x, a coordinate which is p.
And you can often describe a classical system
if you know the phase space distribution,
if you know the probability that a particle has
position x, and momentum p.
So all of this is about phase space densities.
In a way, let me just kind of-- probability of x and p.
Of course, writing it down you immediately see the problem.
In quantum mechanics, you cannot measure x and p simultaneously.
These are non-commuting variables.
So therefore, what happens is if you now define a phase space
function, which is done in quantum mechanics textbooks,
you can actually do it in three different ways.
And the three different ways are can q, p, and w.
The definition-- I don't want to go
through the mathematical subtleties.
The definition of those functions
involves the operator definition, a and a dega,
or which is equivalent, the x and p operator.
If you define something in units of x and p, or a and a dega,
you can have a product which is fully
symmetric in the ordering of x p, which
is anti-normal or normal.
So in other words, the order matters,
and you have three choices.
In an operator product you can have symmetric ordering, which
means not xp, but xp plus px.
Then it's over two that's symmetric.
You can have an ordering which is called normal,
and one which is called anti-normal.
Ordering of operators in the operator-based definition.

Of course, if we have three choices,
you would say which one is the best?
Which one is the winner?
But the fact is, all three have their advantages
and disadvantages.
So they all have pluses and minuses.
The reason why I picked for the course, q of alpha,
is that it's a real possibility, it's always positive.
It is a diagonal matrix element of a statistical operator,
and this has to be positive.
So it's a probability.
The other guys, p of alpha, can be positive or negative,
and also w of alpha can be positive and negative.
So if you use the p-alpha distribution,
statistical operator can be written like this.
And as a result, the coherent state is now not this Gaussian.
It doesn't have this Gaussian distribution
as a quasi probability, it's what you want.
Well, maybe some of you wanted to see it right away,
it's a delta function.
The probability of the coherent state alpha
has delta function peak at alpha, which is sort of nice.
And the number state is not sort of a ring of a finite radius.
I just mentioned to you the number state.
You would not easily expect, the energy is sharp,
the square of the energies electric field
shouldn't it be charged sharp.
And indeed, it is sharp.
It's actually worse than a delta function,
it's a derivative of a delta function.

But at least here, in the probability p,
which is also called the Klauber pseudo shon p representation,
you get sort of that delta function,
which may be very natural for certain purposes.
Is this the [INAUDIBLE] and you can express one
in front of the other?
Is that unique?
It is unique by some symmetry choice here.
So that's the advantage of it, that by-- I'm not
[INAUDIBLE] the mathematics, I'm not giving you the definition,
but the way how it's defined, it is unique.
It can be written in such a way, and the way
how the quasi probabilities, p-- the p presentation is defined.
You get a delta function at the coherent state.
And finally, w stands for Wigner distribution.

And the Wigner distribution is something
you actually find in most quantum mechanics textbooks.
The q and p distribution are more common in quantum optics,
but the Wigner distribution has the advantage
that the projection on the x and y-axis
are indeed c of x square, c of p square.
So you get actually the x-wave function
and the p-wave function.
So now as a full disclaimer, if you
want the electric field, which is the momentum of the harmonic
oscillator, which is the electromagnetic field, what you
really want to project is the w function, because for the w
function, the Wigner function, the protection
is exactly the momentum distribution
in the electric field.
So the Wigner distribution is closest to the classic phase
space distribution.
As close as you come without violating commutators.

But of course, it has the disadvantage
that it has negative values, and try
to explain to the next neighbor what is a negative probability.
Some people actually see negative probabilities
playing out the non-classical character.
And I know in our field, in AM or physics,
a few years ago during my lifetime,
there were really attempts to use quantum state tomography
and measure for the first time in negative Wigner
distribution, and show that to the world.
So that meant something to a lot of people.
Anyway, you can read about it in quantum physics textbooks.
All I want to know is to know about it,
but then also just relax.
In the bigger picture, all this three distributions
are the same.
It's more sort of on the level of whether something
is a delta function, or it has unity.
So on that small scale it matters,
but if you map out something on a bigger scale,
they are all related to each other.
And for the rest of today and the next class,
when I show you those phase space distribution, when
I see I project onto the vertical axis
to get the electric field, I'm not
completely rigorous which of the three functions
I've really chosen.