Okay, hello. We're just starting to discuss decoherence and let me just remind you of

a few things. So we have some system plus some large environment which we usually

call a bar and there is some interaction between them. In the end we will be

interested in what happens with the system and typically we will look at

some interference effect and we will see how this interference effect is then

destroyed by the interaction of the environment. And so the first example

that I gave. So the first example I gave was the simplest possible example you can

have for the system which is just to have a spin and for example if the

Hamiltonian of the spin would just describe some energy h per omega to the up state

then in the absence of coupling to the bar this is just a spin in a magnetic

field and so if you prepare it in a superposition of up and down which means

that it points into the x direction it will just precess in the x-ray plane.

And so if you then plot a sigma x as a function of time you would see these

nice cosine type oscillations. Now the question is what happens if you couple it to a

bath? Everything will depend on how exactly does the interaction look like

and how exactly does the dynamics of the bath look like. The simplest kind of

interaction is this Q and D type of interaction which we have already

discussed for the measurements where the interaction itself would commute with this

Hamiltonian. So if you want to see an explicit interaction Hamiltonian this Q and D

type of interaction would look like this for this two level system. So the up state

would couple to one particular operator of the bath let's call it the up and the

down state would couple to another operator of the bath let's call it the down. So that would be an

example of a Q and D interaction. And so what happens then is that depending on

whether I'm in the up or the down state with respect to the spin the bath will

be both differently because it is subject not only to its usual bath Hamiltonian

but also to this additional the up or the down and so two different bath states

develop this is exactly the situation we already discussed for the measurement

you would have two states Q up and Q down. So the state after some time of system plus bath

would look like this if I suppose I have an equal position initially but this

doesn't really matter so much. I would have down times Q down of D this is the

state of the bath given that my spin is in the state down plus there is up times

up times D up of T. And then there is the bare evolution which in this case is simply oscillation at the frequency higher.

Now if you go ahead and now calculate for example the expectation value of sigma x

you will find that the expectation value is modified it's modified precisely by

the overlap of these two bath states. So that is what we wrote down. So in particular in

the simple situation where this overlap is purely real value which need not

always be the case but simpler in that case to see what happens the overlap just

becomes a pre-factor in front of our cosine type oscillations. So that would be

chi down of T and chi down of T. Or more generally even if I were to say calculate sigma y again the same overlap

would appear and so you can express that by calculating the reduced density

matrix of our system that is our spin in this case. And that's write down say the element up down

which is one of the two half diagonal elements and then it happens to be precisely

this overlap in the opposite order so there is chi up of T to the right and chi down of T to the left if you work it out

times the bare time evolution the evolution of the half diagonal element in the absence of the graph.

So that's called a zero bare time evolution which would just be an oscillation at this frequency.

Okay now if I plot this, plot the expectation value of sigma x for example the typical situation will be that as time progresses

the two buffs diverge more and more from each other so the overlap gets reduced more and more.

And so typically I have just some decay and the envelope here is determined by precisely the time evolution of the overlap.

This need not necessarily go down to zero that depends on all the detailed properties of the bar.

It could also show revivals at some point in time so there are many different possibilities depending on the physical situation