So, I hope you are still motivated to learn these things.

I will continue essentially directly on the subject today.

So yesterday I told you how these optical forces can be used to engineer interaction between these membranes and the atoms in the optical lattice.

And what I want to do first now is I want to give you again without doing calculations the simple physical picture of how this interaction arises.

And then I will discuss one first application of this system which is to use the atoms as a coolant for the membrane.

So, to cool the membrane by its interaction with laser coolant.

I will then show you an experiment that we implemented this and observed this to room temperature setup to the level of militech.

And then I will take a break with the slides essentially and discuss a little bit what the essential parameters are that describe this topic system.

And it will turn out that the cooperativity which we defined plays a very important role there.

And based on an analysis of these cooperativities one can show that cooling with the atoms is in some sense more powerful than cooling by a clarity of mechanical techniques or cold denting if you just have the membrane inside the clarity.

This statement is true for certain regimes of the clarity and membrane system where you cannot ground they to the standard of mechanical techniques for example because it goes side to side with the solution and so on.

But you can still do it with the atoms and I think that is a very interesting result because it tells you something about the possibility or the power of such coherent controller if you want as opposed to a classical detector and controller.

And after this discussion I will then tell you a bit how we can extend this system to cover to the internal states of the atoms and what new features arise if you deal with internal states.

And some people asked me yesterday to explain again why one can connect to Eugene Bolzic's lecture who is also discussing this topic and some people asked whether I could explain again how one can describe such a spin system as an impossibility of a dynamic mass or I will say a two-mph power of performance request.

Okay so this is the program for this last lecture. So let's again have a look at the system. I remind you we have a single side clarity with such a membrane in the middle which is a standard of two-mechanical system which has been used in many experiments.

You might as well just think of the membrane as a very good mirror and leave out this left part here and then it's the standard generic system with the moving end here that will work equally well. We just use the membrane because it's a very high quality attempt to oscillate.

The single sided cavity is coupled with the laser that comes in and across the single sided. All the light comes back and you get a nicely modulated standing wave here in which you can track the atoms. And because this standing wave is what? It's a laser standing wave.

It extends over very long distances. You can reshape it with lenses and all that. The atoms actually don't have to sit very close to this system but they sit in a separate vacuum chamber more than a meter away. Actually in our current experiments we even have optical fiber in between.

So these are really remote locations. And I think this is also kind of an interesting conceptual point. What can you do with a quantum system in such a light beam to control the remote system that seems to be completely in place and apparently you can cool it and control it.

Which I find interesting. So this long distance nature of the coupling and modularity of the two setups is of course also experimentally a very nice feature because you can independently optimize the systems and exchange one thing here without only this one.

So how does the coupling work? I will now give you a very simple picture in terms of optical diver forces and radiation pressure force in the membrane as well as this back action of atoms on the light field which we discussed two lectures ago.

So remember when an atom is moving in such a standing way it can print its motion on the laser beam that has created a phase distance. So how does the membrane affect the atomic motion? We looked at this yesterday.

So if the membrane is vibrating it shifts the frequency of this cavity and in the single sided cavity which is resonantly coupled the main effect is to have a phase shift of the reflected beam which then displaces this interference pattern.

And this creates the dipole force acting on each atom and because the lattice rays are ideally identical you have such a force on each atom and therefore you couple to the atomic center of mass.

And one thing we can emphasize in this context is that this coupling somehow scales with the finesse of the cavity. Why is this so? Imagine you don't have a cavity. Then it's very easy to calculate the phase shift of the reflected beam if the membrane is moving.

It's simply related to the displacement of the membrane and the K vector of the light. So in order to have a phase shift of 2 pi here you have to move the membrane by half a wavelength if there is no cavity.

On the other hand if the membrane is inside the cavity then already a tiny displacement is sufficient to shift the cavity by its line width in which case you would also get a 2 pi phase shift.

And the factor that connects these two situations is the finesse. Simply because the light bounces back many times the displacement that is necessary to shift this lattice by one period is smaller by the cavity finesse in the case where you use the cavity.

So in some sense the cavity acts as a lever that transduces this small motion of the membrane into a much bigger displacement here which enhances the coupling.

Okay, how does the back actually work? So how do the atoms influence the membrane? Well first one has to recall that there is a mean radiation pressure force on such a membrane inside the cavity on the slope here.

The light field exerts radiation pressure and pushes the membrane to the left. That's the steady state situation if the atoms sit at least spatially in the membrane.

So now let's assume the atoms are displaced. What happens? Well they experience this restoring optical dipole force. Each atom experiences this dipole force and as a result we learn there is a photon redistribution

between the light wave that comes in and the light wave that goes out with their opposite momentum. And so for a force on the left you absorb photons that come in and directly reflect them out so that they actually don't ever enter this cavity system.

So this means the power of the laser beam that has gone through the atomic ensemble is now reduced for a force that points in this direction. And now you can imagine as the atoms move back and forth there is a power motivation on this beam.

What will this cause? Well this power motivation will be also, this power and this cavity is enhanced by the finesse and as a result this power motivation is also enhanced and leads actually to a quite strong modulation of the radiation pressure force on the membrane.

So this is how the atoms act back on this membrane. And it's important to realize that again the finesse of the cavity enhances the effect because this power motivation as the light in the cavity is enhanced by the cavity finesse which is the factor by which the circulating power is higher than the input power.

So in both ways of interaction this cavity helps us to enhance the interaction strength which I find remarkable because one of the systems is not inside this cavity.

You know if you place both systems inside the cavity you would say kind of naturally yes somehow the cavity finesse helps you to get a stronger circulating power but it turns out you can also gain by just placing one of these systems inside the cavity because the other one is a remote system.

So if you do the derivation of the Hamiltonian which we did yesterday for this interaction then you'll find that it has a simple form. As a result you can see there are forces proportional to the displacement so this gives a linear interaction between the membrane displacement and the atomic displacement.

This coupling constant we also calculated and I showed that it's proportional to the optomechanical coupling and to the potential C-9 atoms.

If one plugs in all numbers such as the formula for the optomechanical coupling, the membrane system and so on one can show that this coupling strength also has this formula. So it scales with the reflectivity of the membrane as one might expect.

The characteristic frequency scale is the oscillation frequency. That's also quite natural. That's the frequency scale of these two oscillators which we assumed to be linearly resonant.

This we can achieve by tuning the traffic between the lattice.

And then there is the factor which is the square root of the mass ratio of the two systems. So what's in here is n times the atom number times the mass of a single atom.

So that's the mass of the entire atomic ensemble or if you want the mass of the center of mass motion. And it's divided by the effective mass of this membrane.

Now you can imagine this membrane is much more naturalistic of course than the other atoms. So this is a very small factor. In our case it's 10 to the minus 10. Even if you take the square root it's still 10 to the minus 5.

So this is very small and it can be understood as an impedance mismatch of the two oscillators.

So if you have two oscillators that are coupled and they have the same frequency but very different masses they do not very effectively couple.

Think about two pendula, one with a huge mass, the other one with a tiny mass and the spring in between. Even though they may have the same resonance frequency they don't couple very efficiently.

Same thing happens here. But the cavity finesse which can be high, you know, 1000s, 10000s, 10000s or something like that, allows you to compensate for this impedance mismatch.