So good morning, let's start. Hopefully others will come but I will start now. I will discuss

the home task in the end of the lecture. Hopefully I will get some more solutions. Yeah, so at

the last lecture we discussed discrete variable quantum optics and the beam splitter introduced

the correlation functions and in particular considered G2 normalized second order correlation

function and its measurement. Today we will continue discrete variable quantum optics

and then I will also discuss with you a couple of other experiments with beam splitters because

as long as we learn how to deal with beam splitters there is plenty of things and how

to deal with loss as well. Hopefully you got the idea. So now let's calculate G2 which

is equal to not normalized G2 divided by mean number of photons squared which is normally

ordered squared mean number of photons. Yeah, let's calculate this value for different states.

We did it in classical optics. We showed that for coherent state it's one, for thermal state

it's two and then I told you that in principle there are also cases where G2 is less than

one. This is called anti-bunching but it is not described by classical optics. So let's

do this calculation in the quantum formalism. First coherent state alpha. It's very easy.

What is the mean number of photons? We know it's alpha squared modulus. Alpha is just

a complex number. It denotes the state and also its amplitude. So n is alpha squared

and then also what is this normally ordered n squared? We have to write the normally ordered

photon number operator and sandwich it between the coherent states. So we have alpha and

then a dagger squared a squared alpha. But this is very easy because you know that the

coherent state is the eigenstate for the photon annihilation operator a and it means that

as a acts on alpha each time you just have a factor of alpha and alpha bra vector is

the eigenstate for a dagger and the same happens here in this part so alpha complex conjugated

comes out of this. So easily we get alpha to the 4. Yeah, this is clear. And by dividing

one by the other squared you get of course that g2 is 1. And actually we can write more.

We can write that n to the k normally ordered squared is alpha a dagger to the power k a

to the power k alpha and the same thing will happen so this will bring you alpha to the

power k this will bring you alpha complex conjugated to the power k which gives you

alpha to the power 2k and k-th order correlation function will be also 1. It's clear with this

so absolutely strictly without any assumptions quantum theory gives us that normally ordered

normalized correlation function for of order k for the coherent state is 1. Now let's consider

another pure state before I come to thermal state I want to consider some other pure state

thermal state is a mixed state as I told you. So for a pure state m what is g k in the general

form. So let's calculate first not normalized correlation function which is I have to sandwich

k, k sandwich over the state n construction n to the power k normally ordered but at the

last lecture we proved some identity that n to the power k this is an operator of course

this operator can be written in the form n times n minus 1 times and so on. So this factorial

moment n minus k plus 1 and this is sandwiched between the folk states but here k 1 and so

on they are just numbers and it means that the folk state is an eigenstate also for all

these operators and the eigenvalues of course are these n minus k plus 1 and so we get the

simple construction n times n minus 1 and so on n minus k plus 1 and here n is no more

an operator it is just the number of the state. This has to be divided by the mean photon

numbers to the power k but the mean photon number for the folk state of course is just

n it's just the number of the state for the same reason and then we calculate easily g

of the order k is just this construction n, n minus 1 and so on n minus k plus 1 divided

by n to the power k these are key brackets you divide by n to the power k obviously this

is less than 1 right because you can you can remove this n so this will be n to the k minus

1 and each here each bracket is less than n and we see that this is less than 1 so we

have derived anti-bunching we have derived the fact that for a photon number state any

order correlation function is less than 1 let's calculate g2 and it's easy you just

have to stop after second bracket so this is just n, n minus 1 divided by n to the power