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Hi again everybody. So the first thing is apology. Some of you probably saw the schedule and were expecting a little bit of a go here, but you don't have to say too much.

So we switched on extra times in part because the things that I want to talk about today are just all experimental demonstrations of things that Yande has developed in the first hour of his talk.

And we've got to things like conditional measurements. We haven't gotten to do those yet. So I think it makes sense to tie these two things together.

I have an announcement about my own lecture slides. Sometime tonight I'll repost all three lectures online. And one of the things you'll see, get with us in one or two, there's a set of slides I have to show here which are additional readings.

And I thought I'd just point out there what are interesting things to read for further reading on the topics we've talked about. So if you want to do anything more then let me stop that for now.

Okay, so what I want to talk about today, I really want to get on to the experiments that we've done that are optomechanical by intent in gravitation wave detectors.

But before I move on to the optomechanics, I do want to complete my discussion on squeezing. And one of the things that led to some confusion I thought what I would do is I'd take a step back and really talk about how the non-homotopical interaction actually happens in the OCO.

And this is something that I showed in Hamiltonia for last time in this form here where I just showed that the complete Hamiltonia for this chi-2 nonlinear interaction in a nonlinear material gives you the creation of two, in this case, an A1 and an A2 are simply just for us our infrared photon and our twice as energetic mean photon.

Now in this process what we do is we actually, now some of you saw already coming that we pump the OCO with our green field. So we call that the pump and we simply let that go to being a very strong coherent state. That's the pump field.

So once you do that inside this Hamiltonian, then you can rewrite it as just this. You just replace your A2s with the amplitude of the pump field. And now with this Hamiltonian you can actually simply apply it to a mode.

In this case we applied to a vacuum mode. So you can actually do the evolution of the vacuum mode. And you can see very quickly from this form here that you get the squeeze operator.

So that's really how the squeezing is happening. And I want to really emphasize one other thing in here which is that in this case it is a particular vacuum mode at the frequency omega 1, at the frequency of these creation photons here.

Now what's happening inside the OCO itself, you have to compare it to Gauss-Leroy. This is just the cartoon version of what these waves tell us. So we have this strong pump field. We don't really care about it. It comes right out of our operators here.

And then we see it with a field. In our case this C field would be a vacuum state. But many people are confused about that. In the experiment you actually can't really align a vacuum beam.

So what we do in reality is we see the OCO with a coherent field. And then eventually after we've aligned everything we turn it off. So you can just think of it as a coherent field coming in that eventually you attenuate to have zero photons.

And now indeed as this operator predicts right here, you get two red photons here. This is a four-wave mixing process. You have these two waves and these two waves out. And energy is indeed conserved.

And now if you look at this in the phase-based picture, what you see is here you have some carrier field. And now you have these two. And this carrier field may be just some local oscillator that you use to make the measurement.

Now remember the other thing that's very important about vacuum states is you really can't measure them without beating them with a coherent field. And so that's the local oscillator field you use.

And you have these two side bands. So this picture is already in the rotating frame of the local oscillator. And now these two side bands are counter-rotating at some frequency omega.

That's the frequency at which we will observe the squeezing. And you notice indeed that when you actually time-adjust these rotating side bands that are tied over to our face, you get a total noise that's minimized along this axis.

And these two things sum up along this axis and they cancel along this axis. That's really the picture we should carry of what you're doing in the experiment.

May I just ask you a question on that? Because, okay, so I imagine that if you're in vacuum, you cannot choose a particular mode that's on vacuum.

No, that's not true.

So you're going to do a pump? No?

No. You actually very much choose precisely the mode. All possible modes are there. You have chosen the mode determined by your pump right here.

You end by both in spatially and in frequency. So that's really part of why, when I show you my squeezer, let me just show you the squeezer.

All of this stuff here is simply just to select that mode. I want to select of all possible vacuum modes the mode in which my interferometer is operating.

And that's why this laser on the squeezer is actually phased off to the interferometer mode. I could just assume, remove all this, and if I knew how to pump my own, you know, just with a laser field that is exactly the same as the interferometer field, I wouldn't need any of this stuff.

So I am selecting very specifically the mode of the interferometer. That includes spatial modes.

So you'll see in a minute when I talk about losses, we have to actually make the spatial mode of the squeeze gate be the same as the spatial mode of the interferometer.

It has to go through all the same lensing.

Okay. But if you could give me a number, please. Like how far does this specific mode extend spatially?

So it's really, in the transverse direction to the propagation, it has the same concurrence as the laser field. Right? So that doesn't matter.

In the transverse direction, it has the same spatial extent as the mode that's resonant in the OPL. So the OPL is a cavity. And so that selects that mode. Now that cavity does not have the same mode as my interferometer. That cavity is centimeters big, my interferometer is kilometers big.

So we do the exact same thing as you do coming out of the laser. When you come out of the laser, you have a small nice loop of beam. It expands out to be a 12 centimeter diameter in 4 kilometer long cavity. That's not the lensing.

So it has all the same properties. You cannot think of the vacuum as anything different than an electromagnetic field. It is what it is. Right?

So just think of it as, really if you have confusions, think of it the way we do in our experiment. We start with a real laser beam and we block it. Okay.

Alright. So that's the generation of the squeeze tape. I just want to show you a measurement that we did.

Previous slide, the one which you have is coming clean. Sure. I guess this is A2 goes to R2P based on the R2P. Oh, yeah. A2 goes to R2P. Sorry. Yes.

And also this fingerprint does not match the opera speaker. This is just a single mode of squeezing. Yeah. Look, if I wanted to do the quadrature squeezing, I would have to do transformations into the quadrature space, which I haven't done.

But really the analogy between two mode squeezing and quadrature squeezing is straightforward. So I don't think you should get hung up on that.

Alright. So here's what we did. Here's the experiment. Here is my squeeze generator. I've sent my squeeze tape into the interferometer.

And the thing that I didn't get to last time is to show you the actual results. And that's what these are.

So what we did was we did this experiment in the initial LIGO detector. Immediately between the time the initial LIGO was completed and the advanced LIGO was begun, we kept one of the LIGO detectors alive for a few extra months.

So the red curve is the sensitivity that the detector had at the time we made this measurement, and that's without any squeeze injection. And then the blue curve is the improved signal-to-noise ratio or the reduction in the shock noise level once we injected the squeezing.

And you can see that you basically can see this reduction kind of all the way down to here to about 150 Hz. And that's because below 150 Hz we were not limited by shock noise. So we obviously weren't going to see any improvement in the quantum optical noise.

And now the other thing that you should notice is that this improvement is about 2 dB, which is 25% improvement.

So the question that then comes up, so this is actually a very important measurement in our field, because one of the things that we have to prove here is not just that you can squeeze, and we've been squeezing for a while, but that you can squeeze without doing any further degradation of the interferometer in this region.

So we have to keep all the couplings down. Now, why 2 dB or 25%? I mean our squeezer, basically when we measure it on the table, we make 12, 14 dB squeezer which we can measure.