Okay, so let's start our last lecture this year and I'm going to finish what I started

during the previous lecture that is continuous variables. I explained you a lot about discrete

variables in quantum optics but we have just approached the idea of continuous variables.

Yeah, thank you for the home task. And it will be short because the most part of this lecture

I hope to dedicate to polarization in quantum optics. First continuous variables. Continuous

variables in quantum optics are quadratures. Of course there are also position and momentum

but we will only deal with these quadratures x1 and x2. They do not commute. Yeah, I think

it's i as far as I remember. It depends on the definition of course. Please take a seat.

Take a chocolate as well. And so what else? First of all what we can calculate in quantum

optics about some state. What features of a state. Of course we can calculate the mean

values and the variances of these quadratures and you should be able to calculate it. But

I want to introduce some variables. Functions of these quadratures. And as the variables

we will look at three kinds of quasi probabilities. As I told you because the quadratures do not

commute well I will write it just non-zero. It means that I cannot introduce the probability

that one quadrature takes the value x1 and the other quadrature takes the value x2. Because

if these values do not commute then there is uncertainty relation and then they just

cannot be defined simultaneously. So this probability if I introduce it in some possible

way it is going to be to have some strange features. And the strange feature we will

have is the negativity or singularity. So there will be three quasi probabilities. The

Glauber-Sudarshan function which we will start with. How it is introduced. We looked at coherent

states at one of the lectures. So alpha coherent state. And in terms of these coherent states

one can introduce some density matrix. Again we spoke about density matrix rho and the

density matrix can be written in some representation. So for instance we wrote the density matrix

of the thermal state you remember in the representation of coherent states. But now we will write the

density matrix in the representation over, well previously I said I meant Fock states.

But now we will write the density matrix in the representation over coherent states. And

it will look like this. So it will be integral. I will instead of alpha I will use the variable

z. z is the complex variable. It will have real part plus I imaginary part. And here

I will write the two dimensional integral over this complex variable. And the density

matrix P will be a function of this z. So it is a function of complex variable. So if

you want it should be the double integral. And here of course the projector. So the projector

on a coherent state with the amplitude z. It means that I am considering here a complex

plane z prime, z double prime, so real and imaginary part. Here P of z. And I am going

to characterize by this function P the properties of any state. Because this is in this notation

it is the density matrix. This is the density matrix of a state. This P of z. Well I can

write that z prime corresponds to x1. I can write that z double prime corresponds to x2.

It's the same. And this function is called Glauber-Sudarshan function. Or quasi probability

or P function. And it is maybe the most important function in quantum optics. Because it defines

according to the form of this function for any state one can say whether the state is

classical or non-classical. So the definition which we will discuss probably later in detail

in a lecture about non-classical light. That light is non-classical if this P function

takes singular or negative values. And otherwise it's classical light. So this is the definition

of non-classical and classical light. But we will now discuss a little bit how close

it is to probability or quasi probability. For instance, can you understand now if I

have a state alpha, coherent state alpha, what will be its P function? Just a guess.

To have here, well to have a state alpha, I need to have here the projector on the alpha.

The density matrix of a pure state is a projector on this pure state. So how should P of z look

to have here this state? Delta function, exactly. So P of z will be delta function. I put an

index 2 because it's a two-dimensional delta function of z minus alpha. So it means that

for a coherent state I have somewhere some complex number alpha and this P has singularity