Okay, so today I will focus on two nonlinear effects that produce non-classical light,

that's parametric down conversion and four-wave mixing, and we will derive quite rigorously

the generation of photon pairs and squeeze light.

So as I, at the last lecture I was writing that nonlinear polarization, in the general

case polarization depends on the electric field nonlinearly, so it's chi 1, this is

a tensor, E plus chi, ah, okay, there is epsilon naught, epsilon naught, plus epsilon naught

chi 2 E times E plus chi 3 E E E.

So we'll use today the first two nonlinear terms, this term and this term, and I will

start with chi 2 and I will describe parametric down conversion.

As we already mentioned last time, I think it's better this way, as we already mentioned

parametric down conversion is a process that happens when you shine strong, rather strong

laser beam into a crystal with chi 2 and the crystal has a length L and suppose that there

is a pump with a diameter or waist A, so the pump comes from here and I deliberately highlighted

only part in the crystal that is covered by the pump.

So the pump is propagating this way and we are going to describe how on the right some

photon pairs appear, or just something appears, because the diagram we showed last time, we

drew last time is this omega or symbolically I can write here 2 omega and nothing else

is sent into the crystal and it's possible that omega and omega 2 photons or two beams

appear or actually beams with omega 1 and omega 2, different frequencies, there is no

real level here.

So how to describe this process?

We have to write down the energy of this area in the crystal that has both nonlinearity

and the pump and so there is this nonlinear polarization and the energy from electrodynamics,

you know that in the dipole approximation and the energy we will call Hamiltonian because

the quantum, it will be the quantum expression for the energy, the Hamiltonian, will, well

first, first let's just write that this is the energy and this is the product of the

dipole moment of the matter times the field with a minus sign.

So this expression hopefully you know, the energy in the dipole approximation is just

the product of the dipole moment of whatever has a dipole moment and the field and this

will be enough to describe the effect because this energy will then be called the Hamiltonian.

What is the dipole moment?

A polarization by definition is the dipole moment per unit volume.

So the d, d can be described by the integral over the volume where, which we consider and

here this is this p of r d3r.

So we integrate polarization and here we will consider only nonlinear polarization over this

volume.

So what is v?

v is just exactly what I highlighted here.

It's the length l and in the transverse direction it's a, well it's a circle of course, but

we'll see.

I will introduce a frame of reference.

The z coordinate will be in this direction then there will be x and y coordinates in

this direction.

Then I can write that this energy, I will put the integral, I will put the integral

outside of this expression and then I will not bother with constants here because I want

to just derive the principle things.

So I will omit the minus sign and I will omit some number coefficients, epsilon not definitely,

I don't like it.

So the Hamiltonian will be an integral over this volume v and then the polarization I

will take p second order nonlinear polarization.