All right, I guess it's that time again.

So last lecture, last pick of my two lectures for today.

Hopefully we all make it through.

But again, don't be alarmed by the stuff already on the board.

The notes for the lecture I'm giving today are on the website already.

So you have a copy of this on the website.

So what I want to do today is a quick recap of what we did last lecture.

This back-action evading scheme where you just simply draw an often-accountable value with two tones.

You can measure just one account of the other chart.

I want to go through the story in some detail.

How do you get conditional squeezing from this?

So this connects to your base last lecture where you kind of broke down a conditional master equation,

a stochastic equation for how the density matrix evolves.

I'll try to give you a sense of how that actually comes from the formalism we've already developed.

It won't be completely rigorous, but it won't be completely not rigorous.

In that gray area, you can put those two elements.

And then hopefully, if there's time, I'll also try to connect this story to something that Yung May talked about,

which is the idea of coherent feedback.

So as opposed to doing something really complicated, analyzing the results of the measurement,

figuring out how to produce the optimal force based on that measurement record to produce real squeezing,

is there some way we can get the system to do all of that by itself?

And what we're doing is actually a very, very simple tweak to this scheme that lets that occur.

The system does the measurement and the feedback itself and automatically generates mechanical squeezing.

So we like to think of that as reservoir engineering as opposed to coherent feedback, getting dissipation to do something useful for you.

And I'll even hopefully show you that this has now been implemented in two very recent experiments.

Okay, so let's start with the recap of what we talked about last class, our last lecture.

We have our standard optical mechanical cavity.

We're working in the good cavity limit where the mechanical frequency is much bigger than kappa.

We're going to drive both the mechanical sidebands with equal magnitude.

So as usual, we write our cavity field as a classical part.

The deoperator describes everything else.

This classical part has two tones, a cavity frequency plus or minus the mechanical frequency.

So as usual, we linearize our optomechanical interaction.

So this is what the Hamiltonian looks like before I've gone into any kind of interaction picture.

So my mechanical position is b plus b dagger.

The cavity photon number a dagger a, that I linearize.

So I have my classical part alpha, and I've taken this alpha naught to be real.

And then I have my, you know, this cavity operator in this displaced frame.

Okay, so then the next thing we do, and again here are my mechanical frequency, my cavity frequency.

I now go into an interaction picture.

And in this interaction picture, b and b dagger will pick up oscillating exponentials.

d will pick up an oscillating exponential, but that will cancel this.

And so then what do I do? I throw away all of the terms in this interaction picture

that are fast oscillating at plus or minus twice the mechanical frequency.

And what am I left with? I'm left with this very simple QND Hamiltonian.

So now b and d mean something different than in this Hamiltonian.

We've gone into the interaction picture.

So this b plus b dagger is no longer position.

It's one of the mechanical quadratures.

It describes the cosine component of the mechanical motion.