Okay, so we're just discussing the quantum states of the electromagnetic field, and so

that for us just means the quantum states of the harmonic oscillator in the field mode.

And so, as a nice description for this, we just introduced the Biegner density last time,

and the Biegner density is just a sort of Fourier transform of the density matrix.

Okay, so you see that X is sort of the center coordinate, and Y is the difference coordinate.

You keep the center coordinate that becomes the coordinate at which the Biegner density

is evaluated, and you do Fourier transform with respect to the difference coordinate,

and that gives a Fourier variable namely the momentum.

And we discussed several cases, for example we discussed the ground state for which the

Biegner density is just a two-dimensional Gaussian in the phase space plane of X and P.

The coherent state is just a displaced ground state, so it's just displaced two-dimensional

Gaussian in this phase space plane.

We discussed the squeezed states where the contour lines of the Biegner density would

look like ellipses.

So that, for example, would be a squeezed state.

And now at the end of last lecture we discussed ways of actually being able to measure the

Biegner density, and the one method I discussed was so-called quantum state tomography, where

the idea is that if you're able to say measure the Biegner density, if you're able to measure

the probability density along the X direction, and along the P direction, and along any arbitrary

other direction, then for each of these measurements you essentially obtain the integral of the

Biegner density along the other axis.

So if you measure the probability density as a function of position, you have to integrate

over momentum, for example.

And we found by being able to measure along all arbitrary axes, you could reconstruct

the two-dimensional Biegner density, and this reconstruction is based on the same principles

that I used for reconstructing the density of a body state after you have illuminated

it with X-rays.

So that was quantum state tomography.

I want to end this chapter about the Biegner density by pointing out another nice method

of being able to measure the Biegner density.

And that would become important when afterwards we describe the circle quantum electrodynamics

experiment, where they were able to use this method to actually reconstruct the Biegner

density.

So while this method is based on being able to measure things like X and P and other combinations,

the method I'm going to discuss now is based on being able to measure the probabilities

of having different photon lambdas.

And so what we want to start with is just the expression for the Biegner density at

the origin, because the basic idea would be if we are able to somehow get a measurement

of the Biegner density at the origin, that is at X equals 0 and P equals 0, and we are

able to translate the Biegner density by arbitrary vectors in the XP plane, then of course we

are done because we can't always translate it by arbitrary amounts and simply measure

at the origin and by repeating this measurement we cannot change the Biegner density everywhere.

So let's write down the expression for the Biegner density at the origin that is just

stating this expression by setting X equals 0 and P equals 0.

So that would be the integral over Y of rho Y half and minus Y half.

Now, the next step would be that I want to make contact to the energy eigenstates, that

is the fox states, the photon lambdas states.

And that means I want to expand my density matrix in the fox state basis.

Just to remind you, rho of X and X prime is given if I have a pure state as psi of X,

psi star of X prime, and then if I imagine that instead of a pure state I just have a