So today I want to finish the chapter about Optomechanics and I want to say something

about the general future trends in this field.

So one of the things that people think about is producing some interesting quantum states

and the simplest kind of interesting quantum state would be, beyond the coherent state obviously,

would be the squeezed state.

So the question you can ask, squeezed state of what?

The life or the mechanics?

And actually you can use Optomechanical systems to produce both.

And squeezing of the life here is really very simple, I'll deal with it only briefly.

The idea is that such a cavity with a movable mirror really looks like a non-linear optical

medium.

So the idea is that you get an optical frequency shift that goes like a displacement, but then

again the displacement is provided by the force, by the radiation pressure force.

So at least in the case when you assume that there is an instantaneous reaction, the displacement

goes like the circulating light intensity.

So overall you have an optical frequency shift that goes like the circulating light intensity.

And this is similar to what happens, or this is completely analogous to what would happen

if you had a medium with a refractive index that changes in response to the light intensity.

And such a medium is called a Kerr medium.

And now it is known that if you pass light through such a medium, this can lead to squeezing,

and the idea is very simple.

If you send in a light beam that has fluctuations in intensity, then you can arrange things

such that, for example, if the intensity fluctuates upwards, the optical frequency shifts such

that, for example, the transmission is lowered, and when the intensity is lower, then the

transmission is higher.

So overall there is a tendency to get rid of the intensity fluctuations.

And so that leads to what is known as amplitude squeezing.

Okay so squeezing of the light field via such an optomechanical cavity is really straightforward.

What I want to discuss in some more detail is squeezing of the mechanics.

Now the idea will be to exploit this optical spring effect that we talked about briefly.

That is the fact that the light field can also change the spring constant of the mechanical

motion.

Because if you can do that, you could also change the spring constant as a function of

time, simply by changing the laser intensity, because the optical spring effect, just like

the cooling effect, is proportional to the laser intensity.

So you tune the laser intensity, and thereby you can tune the spring constant of the mechanical

motion.

So if I draw the mechanical potential that would be the parabola belonging to this harmonic

oscillator, I can change the curvature of the parabola in time.

And now you know what happens if I place a ground state wave packet inside this parabolic

potential, and suddenly say I release the potential, that is I decrease the curvature.

What will happen is in the first moment that the wave packet will start to expand, because

there is less pressure, so to speak, but then the width expands and finally contracts again.

And so what you will see after suddenly switching the spring constant is the following.

So let's say we plot the variance of the position, and at some point I switch the spring constant,

in this case I would start expanding first and then I would start contracting, and I

would get oscillations such as this.

So I get breathing oscillations in the width of my wave function, and that is a squeeze

instead.

But now maybe you are not content with the amount of squeezing you get here, and the