Okay, hello everybody. Let's start our first lecture this year. And last time it

was a lecture about polarization in quantum optics and I didn't finish. So

today I will continue about polarization and hopefully tell you the next lecture

which is lecture number nine which is about nonlinear effects creating

non-classical light. So polarization. I finished by introducing a polarized

single photon which had in short notation it was written like alpha h

plus beta v and so h and v are just single photon states polarized

horizontally and vertically and there was normalization condition alpha square

plus beta squared is one and so the pair the cup the pair of numbers alpha and

beta they are complex numbers but they have this restriction and also they

have the feature that the total phase of this state doesn't matter and we can

attribute the phase just to beta not to alpha and it means that we basically

have two real numbers characterizing this polarization qubit. In fact it is a

polarization qubit as I think I mentioned at the very first lecture a

single polarized photon but before explaining how it can be treated on the

Poincare sphere and so on I want to consider once again the subject of

polarization measurement and this is the Stokes measurement because classical and

quantum descriptions of measurement differ a little bit and because we

introduced the Stokes operators let's consider measurement of the Stokes

observables corresponding to the Stokes operators. In quantum mechanics measuring

an observable corresponding operator means that we have to project the state

on eigenstates on the eigenstates of this operator and so we have to find the

eigenstates of the Stokes operator. How do we do it? We consider this class of

states single photon states and let's consider first S1 which I remind you is

a h dagger a h minus a v dagger a v you remember what a h and a v are

annihilation operators in the horizontal and vertical polarization modes and we

just have to solve the equation that S1 psi is some number S1 times psi and if

we substitute all this we obtain the equation a h dagger a h minus a v dagger a

v times alpha h plus beta v is equal to S1 alpha h plus beta v and of course we

have to first let's let's see what happens when we act by this operator on

this state and this is the photon number operator in mode h this is photon

number operator in mode v so when we act by this operator on this state h we just

get the eigen this is the eigenstate of this operator right so the eigenvalue is

one and if we act by this operator on this state the eigenvalue is zero

because in the horizontal mode here there is no photon there are zero

photons so I remind you that this state basically is alpha one photon in h mode

and zero photon in v mode and the other way around for the second state so the

result of acting by this operator on this state will be alpha h because here

the eigenvalue is one minus beta v because of this minus and this has to be

equal to S1 alpha h plus beta v and then we have to write equality separately

for this vector and for this vector because they are orthogonal so we have a

system of equations first equation alpha is alpha minus 1 minus s1 is 0 by by

just taking the part of this inequality relating to h and for the v part we get

the equality beta 1 plus s1 is 0 and this is a system of equations we see

that the first equation has two solutions so either s1 is 1 and then but

then from the second equation we get that beta has to be 0 beta is 0 and we

get the qubit from this we get the qubit well first eigen state of the first

Stokes operator with the eigenvalue 1 is just well we write this vector like this

1 1 0 because beta is 0 means that alpha is 1 so we get this state and the second

case we get from this equation that s1 can be minus 1 and then from this