Okay, so I think we analyzed the linear dynamics of optomechanical systems last time, and so

as a natural next step I want to turn to the nonlinear dynamics.

But as we remain in the classical regime first, but we want to look at the nonlinear dynamics.

In particular we want to look at what happens if a system really becomes unstable.

And we already learned it can become unstable because you cannot only cool such a system,

but you can also switch to the other side where the light field actually puts in extra energy into the mechanical motion.

And so if that over burns the friction, then you can go into a regime of what we call self-induced oscillations.

So that's what we want to discuss.

So just to remind you, the idea was that the full damping rate contains the intrinsic mechanical damping

plus some extra contribution induced by the light field, which is proportional to the intensity of the incoming laser beam.

And this can also be negative depending on the detuning.

If the laser beam is blue detuned, then it submits some extra energy to the mechanical system.

And so if this is negative, and then if even the whole damping rate becomes negative, then you can go into instability.

Because what happens in such a case, again just to remind you, is that if you have any tiny initial fluctuations

to start with, for example thermal fluctuations, then because you have negative damping, these blow up exponentially.

And only when the amplitude of these oscillations becomes large, then this linear analysis is no longer valid.

And that means there will be nonlinear effects, and these nonlinear effects make the amplitude of these oscillations saturate.

So one of the reasonable questions to ask is, what determines this final amplitude of the oscillations?

Well, and obviously, a priori this amplitude will be a function of all the different parameters we have.

The mechanical parameters such as the mass, the frequency, or the mechanical damping.

Also the parameters of the light field, such as for example the detuning, or the damping rate of the photons, kappa.

Or also the strength of the laser drive, because of course if you switch off the laser drive, then the amplitude will be zero.

There will be no such oscillations.

Okay. Now, the first thing you can do is simply simulate the equations that we wrote on last time on a computer,

and convince yourself that first indeed this is true, the scenario happens, it does saturate.

And second, these oscillations for typical parameters are still very much sinusoidal, with only very tiny deviations.

I mean, you are not guaranteed that this looks like a sign of omega t, but there are only tiny deviations.

And furthermore, the frequency of these oscillations is practically identical to the unperturbed eigenfrequency of the mechanical system.

Again, this is not guaranteed, and you can choose parameters such as this, is no longer true.

But for typical parameters, this is true.

So, the ansatz you can make for our purposes is simply that you have the sinusoidal type oscillation,

it will have some offset x bar, which is not necessarily the equilibrium position of the oscillator that is not coupled to the light field,

this can be shifted, and we will discuss this, plus some as yet unknown amplitude, A times cosine omega t, or sine omega t.

Because another observation is actually that the phase of these oscillations is not fixed by anything.

So now, the first thing you can do is assume that the mechanical oscillations have this form, what happens to the light field?

This is an important question.

And even though the mechanical oscillations are very simple of this form, the light field undergoes a more complicated dynamics.

So again, I will plot x versus t, I do have some x bar around which it oscillates.

And then, depending on my detuning, at some position, I will have my optical resonance.

So some position of my burrow or cantilever will correspond to the laser beam in resonance.

So that is actually if x times the optical frequency divided by L, that was the frequency shift, plus the detuning equals zero.

Then you have resonance.

And so now what happens with the light field?

What I want to plot is simply the intensity, that is alpha squared, as a function of time.

And now imagine that everything is completely adiabatic, so the motion is very, very slow compared to the time scales of the electromagnetic field.

Then, whenever you pass through the resonance, the intensity will shoot up.

So that would happen here, and it would happen here, and here, and there, and so on.

And that repeats periodically.

Now, if everything is adiabatic, this would simply be a Lorentzian, and then there would be the next Lorentzian, and so on.

But things are not necessarily adiabatic, and this is important for this purpose.