Okay, let's start because it's getting late. We'll discuss today in the first lecture

Polarization Quantum Optics, which was actually a subject almost one third of my course in

the summer semester, but now I will just recapitalize it a bit shorter and then in the second part

I will start non-linear optical effects.

The task will be explained in the end again because maybe someone comes now and brings

home task.

So polarization, why polarization?

Because there is a very good and deep analogy between different systems in quantum optics

and namely between different so-called two level systems or systems that have, that can

be in the superposition of two states.

For instance, with the atom we already discussed that an atom can have the ground state and

the excited state and the state vector of an atom can be a superposition of the ground

state and of the excited state.

We will write it alpha g plus beta e and of course for the state vector to be normalized

alpha and beta should satisfy the requirement alpha squared plus beta squared is one.

There can be, the atom can be in a mixed state as well, but the mixed state is not interesting.

This quantum superposition is most interesting because it forms a so-called qubit.

We can denote this ground state as zero for instance and this excited state as one.

And here I don't mean the foc states, I just mean that this is the state that represents

the quantum bit zero and the quantum bit one.

Like in classical information theory you have a state or a bit zero and bit one and there

is nothing in between, right?

But in quantum mechanics there is a superposition.

So this pair of numbers alpha and beta are called a qubit.

So the, or the state of the two level system is called a qubit in the general case.

And why polarization optics?

Because a photon, a polarized photon can be, represents a complete analogy.

Just one phrase about something else that has the same structure, you have spin one

half particles.

Yeah, we have spin one half particles for instance an electron.

And this spin one half particle can be in the state of spin up the magnetic field.

So if we have a magnetic field then there is, there are two different states.

There's spin up and spin down in this magnetic field and accordingly a qubit can be also

represented like this alpha spin up state plus beta spin down state.

And this is another qubit, another example of qubit.

So what I'm going to talk about today is the state of a polarized single photon.

So the state will be written like this.

Beta times single photon foc state in the horizontally polarized mode.

And I will soon explain you what it is.

Single photon in the horizontally polarized mode and no photons in the vertically polarized

mode.

So this is single photon foc state plus the other part beta and here vacuum in the horizontally

polarized mode and one in vertically polarized mode.

So, it means that we are considering two modes, polarization modes and in each mode there

can be one photon or no photon.

So this is called a single photon state because we have the same polarization, the same normalization

condition alpha squared plus beta squared is one.

And it means that in total there is just one photon in total.

But it can be distributed between two different modes.

So what are these polarization modes?