Okay, so last time we discussed cooling within the context of quantum theory of quantum mechanical

systems and then a natural question that arises is how do you actually check how cold your

system is.

And that should answer should be that you measure the displacement and you measure the

fluctuations of the displacement and so you can estimate the temperature and then you

see how the temperature goes down as you increase the laser power in order to increase the cooling.

But that brings up another question, namely what about this kind of measurement when you

are very near the quantum ground state because it's not clear a priori that the simple idea

you have of a measurement classically that it is completely non-invasive is still true

near the quantum ground state of the system.

And indeed this really fails at some point and so you have to discuss what happens with

the quantum measurement near the quantum ground state of the system.

And so that will lead us to what is known as the quantum limits for displacement detection.

So the idea we have is that there would be an ideal trajectory and I will start my discussion

in a sort of semi-classical manner.

So first forget about quantum mechanics again, just imagine a classical harmonic oscillator

in thermal equilibrium and we already discussed once the kind of trajectory that you expect

to observe.

That is you expect to observe the oscillations of the eigenfrequency of the harmonic oscillator

but the amplitude will fluctuate over time and so does the phase.

So that would be a typical trajectory and these fluctuations they happen on a time scale

given by the inverse damping rate, that is the damping time.

Now this is not really the trajectory you will observe in your measurement.

For example if you try to measure the displacement by the phase shift of a laser beam and you

club the phase shift as a function of time then you don't precisely see this but instead

there will be some noise on top of the trajectory.

So I want to draw the measured trajectory as opposed to the real intrinsic trajectory.

And you will have some noise, typically some noise at all possible frequencies so you will

get something like this.

And of course the pity is you can't tell what is the noise and what is the signal.

Here of course if you know the signal then you could subject the noise but you don't

know the signal.

So this is the measured trajectory.

So we can say that the measured signal is the intrinsic trajectory plus some noise.

And the question is where does this noise come from?

Now there are interestingly two fundamentally different kinds of sources for this noise.

The first one is the one you think of and which is always there even in the classical

case.

That is simply because your measurement device has some fluctuations.

For example if it is an electrical measurement there will always be voltage fluctuations

and current fluctuations even in thermal equilibrium and they will add to the signal.

And so if the signal is small then these fluctuations will make up an appreciable part of what

you really see as a trajectory.

Also in the case of a displacement measurement with a laser beam you look at the phase shift

of the laser beam but the phase shift will really fluctuate as a function of time because

you try to measure the phase shift in some interferometer.

You have a limited finite number of photons and that makes for a finite error in the estimation

of the phase shift.

So that is what is typically called imprecision of the measurement simply because there is

some extra added noise.