So my question is here are two examples that should begin.

model of the atoms as a simple optical element.

I modeled this thin sheet of atoms that you have

in an optical ledger strap by a thin sheet of

polarized material, and vector sprouts and

activity atoms like is used without.

And this of course makes it easy to include atoms in any optimal set up of interest without having to go through atomic physics calculations if you want.

I gave you the polarizability, I gave you the reflectivity, transmissivity of these optical elements so in principle you can go about them.

Now combine this atomic optical element with other elements such as mirrors or lenses or so and design your atom optomechanical system.

And as a first application of this, let's consider a chemical optomechanical system with atoms like the Hestin's realized for example in the Van Stanford-Hernst group in Berkeley

or the Hestin's group in Zurich where atoms are placed inside the intracal

caly field of the high mass optical cavity.

And these experiments can be understood and especially the coupling strength and these experiments can be calculated by using this model that I introduced yesterday.

And this I will now demonstrate or show you.

The only thing you have to know to understand the system in the beginning is you have to know membrane optomechanics.

And you heard lectures about this membrane in the middle geometry and there you learned that if you place the membrane on the slope of the intracal cavity intensity standing wave

and it moves back and forth it detunes the cavity and this situation is described by the optomechanical Hamiltonian which I show again here with the mechanical part, the cavity part

and then this coupling with G naught the single photon coupling strength.

And this coupling strength for a membrane in the middle has this formula.

So it has this form. So it scales with the reflectivity of the membrane with the frequency of the light divided by a cavity length and the zero point motion.

And now we simply treat this atom as a membrane and we have to plug in its reflectivity and its zero point motion.

And it's not just one atom, it's N atoms. It's an ensemble of N atoms in such an optical lattice trap.

The trap in a harmonic approximation is described by this mechanical trap frequency of the atoms.

So yesterday I briefly mentioned what the reflectivity of such a sheet of atoms is and here it's shown again.

So it is given by the resonant optical depth of this sheet of atoms.

So sigma is the scattering cross section. S is the transverse size of this optical mole field which is taken to be the same as the size of the atomic cloud.

And N is the atom number and so this is the resonant optical depth and as you might intuitively expect if you have a large optical depth, somehow a large reflectivity and this leads to a stronger coupling constant.

Now this resonant optical depth in principle can be very high but it is reduced here because we're working with a multi-tuned line.

And one has this typical reduction factor here which is the line width gamma divided by the detuning from atomic resonance.

And as I emphasized in the previous lectures usually this detuning is at least a hundred lines or maybe even thousands of 10 to the 4 lines.

So as a consequence this is a very small factor. We always work in this multi-tuning limit.

And one has actually to use a decent number of atoms to kind of counterbalance this.

The other thing that enters is the zero point motion here.

For the membrane that's the zero point motion of the membrane. For the atoms one has to be careful.

If I were to act like this it's the zero point motion of the atomic center of mass.

So it's the zero point motion of N atoms, not just of a single atom. Because I also benefit here from the spectra of N in the reflectivity.

I have to pay a price. The atoms as seen as a formula of N particles has N times the mass.

And therefore I have N times N in the denominator and the zero point motion is smaller than for a single atom by one over square root of N.

And so if I plug this in I actually get a correct Huffman constant which I could also have derived from a chemical DQED atomic physics picture.

Yes?

So the equation for the reflectivity it seems like because there are experimental parameters you can control them to be greater than one.

Or is that expressionally valid in certain machines?

Okay, so this specific form of the expression is valid in first of all the limit of large detunement.

They are not bigger than that so this is a small parameter.

And this specific form is valid in the limit of small reflectivity smaller than one.

The more general formula I gave you yesterday where the reflectivity is expressed in terms of this dimensionless parameter, etc.

This is generally valid. So this you could also use for large atomic reflectivities about unity.

Yes?

Are these atoms a gas or something? Are they any gas or solid or whatever?

The atoms are a gas. This is a gas of laser-cooled atoms.