The last time we started discussing optomechanics and in particular we discussed how to measure

the displacement of a mechanical oscillator using the light field.

I already mentioned the fact that if you try to measure the displacement, for example,

by looking at the phase shift of a laser beam that is reflected from the optical cavity,

then the phase shift can be proportional to the displacement and so you would be able

to read out the displacement.

And then we started by looking at the displacement of a single harmonic oscillator in thermal

equilibrium and I briefly discussed that in thermal equilibrium you would expect the

harmonic oscillator to oscillate at its eigenfrequency but to have an amplitude and also a phase

that is slowly varying and it will change on a time scale that is given by the damping

time.

So that is true in thermal equilibrium for a single harmonic oscillator.

I also mentioned already that if you look at the motion of the mirror, then of course

it will be influenced not only by a single fundamental mechanical mode but there will

be many other mechanical modes such as this one and the higher order mechanical modes.

So in truth the position of the mirror will be a superposition of all the normal modes,

all the normal mode coordinates in the mechanical system and therefore what you observe will

look much more complicated than this.

And it is pretty clear that the best thing to deal with this is just to do a Fourier

decomposition because then you can separate the different frequencies of the different

normal modes.

And so that brings us to looking at the Fourier spectrum of the motion and the essential point

about the Fourier spectrum is of course that this is not a perfectly periodic motion, it

slowly fluctuates in amplitude and phase and so if you take the Fourier spectrum of a single

one you will also get a rather fluctuating spectrum and so you have to average over many

ones and it is this average spectrum, the average Fourier spectrum that we want to discuss

now.

And it will turn out that the average Fourier spectrum has a direct connection to the Fourier

transform of the correlator of the potential fluctuations in thermal equilibrium and that

then will also lead us to a connection to the fluctuation dissipation theory that tells

you that the noise at a particular frequency is particularly strong if the system has a

strong response to external perturbations at that frequency.

So this is what we want to discuss today and I want to keep it relatively general because

it applies to all kinds of systems in thermal equilibrium.

Okay so this is a brief theory section because it is sufficiently general and we will deal

with the fluctuation spectrum and the relation to the fluctuation dissipation theory.

So what is the situation we want to consider?

Let's take a trace like this one for a single harmonic oscillator again and take the observation

over some limited time, let's call it tau.

Now we simply want to take the Fourier transform of this.

I want to call this x tilde of omega and now the question is how to define the Fourier

transform.

I want to integrate from zero to tau e to the i omega t times x of t where x of t is

just the single one that I display here.

And now with regard to the normalization of course you know there are different choices

for normalizing a Fourier transform and it will turn out to be very convenient if I enter

one over square root of tau in front.

So let's try to plot the result that you would observe in a single run of the experiment.

X tilde of omega just like any usual Fourier transform will be complex valued.

I want to plot some real quantity and the quantity to plot and also the quantity for