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Good morning everybody. Let me start by telling you a summary of what we did yesterday.

So we talked about the diagonal force.

You can see that was very important. That means polarization.

And we saw that polarization force theory, given here is the volume, and the probability for a two level system.

Let's say an atom is just minus d squared at 6 h bar under the cube.

That works when you shine the two-level atom with a beam that is very far away from you.

That's a really long term.

And you see the probability for the sphere scales with the volume,

and the probability for the atoms can be d squared,

and if you use a relation of d squared with spontaneous emission,

then it means that it will scale like the optical wavelength

for power of three, those analogies.

And then we were writing all the expressions as a function of the polarity

without needing to specify the atomic sphere or every single thing.

So you might put now this polarizable object in cyclic gravity,

and you will be replacing this position where there is a node

at the maximum slope, and I will trap it with some optical prism.

Then I can derive actually the whole physics happening here

by realizing that there will be interactions there

between the dipole and the polarizable object

and the electromagnetic field at the position of the particle.

I just evaluate it by integrating the force,

and I have alpha 4 as the total electromagnetic

evaluated at the position of the particles, squared.

Then the total electromagnetic field here has different contributions.

It has a contribution of the current field,

and that depends now on the evaluated position of the particle,

and it depends on A by eta.

Let me put out the radiation operator,

so that depends on the kinetic wall operator.

Then there is the free part,

that will depend on the position of the particle

and also all the free electromagnetic field modes.

And from these free electromagnetic field modes,

we displaced one of these,

that is the one that will be used as a tweezers.

Therefore, this is the field operator.

Then there was the classical path coming from the strong driving

that is used to convince and start the particle.

This will only depend on the position of the particle.

So, this is the free electric field mode.

And this will only depend on the position of the particles.

Then when we square, we put this here in the square,

we saw that the square of the cavity of this part

leads to the open mechanics,

because we will get a cavity between the position and the cavity modes.

From here we saw that we derived the typical optimist's coupling,

which was minus g naught, e to the power of a, d plus d to the power of a,

because d plus d to the power of a is the rotational relation of forms.