Okay, so we were discussing optomechanics and after the introduction I just want to

focus first on the classical dynamics of these systems.

So that's the usual procedure.

If there is a meaningful classical limit, first discuss the classical equations of motion

and try to discuss the quantum vision.

So I want to start by writing down the classical equations of motion.

We wrote down a part of the quantum mechanical Hamiltonian, but we can really easily write

down the classical equations of motion including all the damping parts, for example, including

the fact that photons decay and that the mechanical oscillations decay.

So, the classical limit simply means for me that we replace the operator x by a classical

amplitude alpha, which is the variable x.

It's more difficult for the light field, but not really that difficult anyway if you know

about coherent states.

So in coherent states, which are the most classical states of the harmonic oscillator,

you would replace the operator a by a complex amplitude alpha that you can identify, say,

with the complex amplitude of the electric field inside the optical mode.

So the equations we are going to write down will be equations in x and the light field

and amplitude alpha.

We start with the mechanics because that is simple enough.

First I write down the equation of motion of a simple thermodynamic oscillator and then

I add to that the radiation pressure force.

So the left hand side describes the free damped harmonic oscillator and then we have the radiation

pressure force and we already discussed that in the quantum limit that was given by the

energy of one photon divided by the length of the cavity times the number of photons.

Now if you are in a highly excited state with a large complex amplitude alpha, then you

can replace a dagger a simply by alpha squared.

So this is the mechanics.

Now for the light mode, let's first try to write down the part that does not depend on

dissipation.

And that is something which we know in principle because we know, for example, what is the

equation of motion for the operator a and that directly translates into the equation

of motion for the complex amplitude alpha of a coherent state.

So that would simply tell you that alpha rotates at a frequency which would be the optical

frequency.

So what is the optical frequency?

Well, in our case it would be the bare optical frequency times this extra factor that takes

into account that we coupled to the mechanics.

So that would incorporate the fact that the light field oscillates at this frequency and

the frequency can change in response to this placement.

Now we have to add two terms and that is taking into account that photons decay and that there

is a laser driving the whole thing.

Now it's not so difficult to write down phenomenologically at least even without derivation a term that

would introduce some damping.

So then you would write alpha dot equals some decay rate times alpha.

So now we already defined a decay rate that was kappa.

Kappa was the decay rate for photons to leak out of the cavity so it would be the decay

rate for the intensity.

Kappa would be the rate at which the intensity alpha squared would decay if you have irradiated

the cavity and then you switch off the light field and then you look at the decay.

So if alpha squared goes like e to the minus kappa t then alpha should go like e to the