So today I want to discuss quantum mechanics and optomechanics.

So far we discussed the classical regime, both the linearized and the nonlinear.

And now I want to talk to quantum mechanics.

And the idea is that along the way we will learn some of the tricks that are needed in general to treat quantum systems.

So let's start with the Hamiltonian, which we already had written down.

First it's the Hamiltonian of the optical mode.

And then the important part is that there is an interaction with the mechanics.

Plus then there is the Hamiltonian of the mechanical mode.

And b dagger b is the number of phonons, whereas a dagger a is the number of photons.

Plus then there will be additional terms that describe, for example, the fact that you have a laser driving the optical mode.

Plus some dissipative terms that describe the fact that photons will leak out of the cavity and photons will be dumped away.

Now what I want to do first is I want to consider the laser drive.

And we will take two steps.

First of all the laser drive is at some fixed laser frequency and I want to eliminate this frequency by going into a rotating frame.

And that will yield a Hamiltonian that is no longer time dependent in contrast to the one I've got.

And we are going to write down now.

And second, once you know the laser drive you know that there is maybe a large light field amplitude inside the cavity.

And then I want to get rid of this light field amplitude simply by considering only the fluctuations around the steady state value.

So these will be the two steps.

First let's look at the Hamiltonian describing the laser drive.

Now what does the laser do?

It just drives the optical cavity.

So for example it tries to create photons inside the optical cavity.

That would be the operator a dagger.

And for this purpose it has to bring along a frequency that matches at least roughly the frequency of the optical photons.

So that would here correspond to time minus i times omega laser times time.

So it's a time dependent driving.

And if you remember the time evolution of the a or a dagger operator then a dagger would evolve according to e to the i plus i times the frequency of the optical modes times the time.

So these somehow match in frequency if they are resonant.

And so this is an important term.

So this of course cannot be a Hamiltonian in itself.

I need to add the Hermitian conjugate to make it Hermitian.

And then in addition there will be some drive amplitude.

So this omega drive is proportional to the laser amplitude.

I could as well have a complex omega drive then I would have omega here and omega star here that just introduces a phase of the laser if I like to have that.

Won't be important for our purposes.

Okay.

But in any case it's a time dependent Hamiltonian.

And so now we want to get rid of this time dependence.

So the words to use are that we want to go into a frame rotating at the laser frequency.

Now I'll explain how this works in detail.

But let me first write down the result because this is completely general.

So what we have in the beginning is a Hamiltonian that contains the operators a and a dagger like the one here.

And by applying a unitary transformation which we will discuss we end up with a Hamiltonian that has the a multiplied by e to the minus i omega l t

that is similar to the result of the Heisenberg equations of motion.

And a dagger correspondingly multiplied by e to the plus i omega l t.

So this is one of the changes but in addition there will be an extra term and I have to subtract h by omega l a dagger a.

So this you can take as a general rule.

It is possible by a unitary transformation to take a Hamiltonian that contains the harmonic oscillator operators a and a dagger and transform it in this way.

And the benefit of this procedure will of course be that if for example a dagger is multiplied by e to the plus i omega l t