I decided to do a little swap with Yanbei against Yanbei just because what I wanted

to tell you about in the second lecture has a lot of overlap with what Yanbei was telling

you in the second half of this talk.

And so hopefully that overlap is a good thing.

I'll try to tell you something about how to think about the standard quantum limit and

the quantum limit on Kishida's position detection in a very, very general way.

It's close to what Yonbe said, but a slightly different spin.

So maybe hearing it a second way will help you understand things better.

Or, worst case, double your confusion.

So we'll see what happens with that.

And the way that I'll tell you is based on an approach that we really started to develop

to understand quantum amplification of microwave signals.

The kind of understanding of measurement of superconducting qubits.

And it's a very, very general way of understanding constraints on measurement by focusing on constraints on noise.

And it can do a really nice way of understanding limits on continuous position detection.

And so for all of the details on the approach that I'll sketch,

there's this review article that I wrote with Florian and others in reviews of modern physics

that has lots and lots of details on this.

Okay, so let me start by doing a quick recap of where we left off at the end of my first lecture.

So I introduced you to the basic theory of the simplest kind of optomechanical system.

We just have a single cavity mode coupled to a single mechanical resonance.

I introduced this fundamental optomechanical coupling.

The photon number of the cavity coupled to the position.

B plus B dagger are mechanics.

And we also talked about all these different transformations.

The first being going into an interaction picture when you drive your cavity.

So if I drive my cavity with a frequency omega L,

we work in an interaction picture where that drive frequency disappears.

In that picture, the equation of motion for the cavity lowering operator has this simple form.

So this delta is the detuning of the laser.

Kappa is the damping rate of the cavity.

So I hadn't written out the explicit coupling to the dissipative baths in this Hamiltonian.

Here's my optomechanical interaction.

And here is, in a sense, the influence of the coupling to the bath of the cavity.

So one part of that is I have some average field that's coming in.

That's my driving field.

So a monochromatic laser drive.

I'm working in a picture where this now becomes tiny dependent.

This is not an operator.

This is just a complex number.

And then on top of that, I have noise that's also entering through this leaking mirror.

So we talked about this noise here is an operator.

We can think of it as operator value, white noise.

OK, so the other picture, the other transformation we talked about,

is we're going to make a displacement transformation.

So I'm going to make a canonical transformation with this a in some complex number

plus a new bosonic lowering operator.

So this is a unitary transformation.

How am I going to pick this alpha?

Well, I want to basically eliminate the classical driving term from this equation.