We've been discussing some of the noise properties in fluency coherence and the example I discussed

just last time was if you have a two-level system and you couple it to a bath, then how

would you calculate how it relaxes from the excited state to the ground state, for example,

and think of an atom coupled to the electromagnetic vacuum and it emits a photon spontaneously.

So the general picture is the one we know.

We have a system coupled to a bath which has infinitely many degrees of freedom.

And in our particular example, the system was really a two-level system.

So we have a spin up denoting the excited state and spin down denoting the ground state

and we have to specify the interaction.

And depending on the interaction, different things can happen.

But the interaction will typically be of the type that you multiply some system operator

with some bath operator.

There might be a sum over such contributions.

And the particular example we wanted to discuss now was when the system operator in question,

the system operator and the interaction really does not commute with the system and the atom.

So for example, in our two-level system, you might choose sigma x or sigma y.

So then the kind of interaction would be sigma x times b.

Sigma x flips the spin, b x on the bath.

And so you would have a situation where, for example, initially your spin was up, then

it was flipped to down, and at the same time there is an excitation created in the bath.

And that then, for example, if you think of the electromagnetic field, is just a photon

that travels away.

Now in particular, we imagine here that the coupling is weak.

So imagine that this b contains an adjustable parameter and we send it to very small values.

And then it turns out that at some point what happens is just simple exponential decay.

So things can be in the end described by weight equations or more precisely master equations.

And our purpose will be to calculate this weight.

Now it turns out that this weight can be completely expressed in terms of the spectrum of the

bath operator b, that is the correlator of the Fourier transform to describe all the

noise properties of b.

So let me first define the spectrum again.

It's just the Fourier transform of the correlator.

And the claim now is that as long as we are in the weak coupling regime, and that's generally

quite hard to define in general what this means, so this has to be discussed in principle

for each system separately.

And as long as we are in the weak coupling regime, all the influences of the bath on

the dynamics of my system can be described once I know this correlator.

And that would include first calculating the relaxation weights, but it will also include,

for example, energy shifts.

There will be an effective shift of the transition frequency, things of the type that were first

discussed in quantum electrodynamics regarding the lamp shift.

Now, last time we just used standard Fermi's golden rule to actually calculate this transition

rate.

And the one interesting conceptual idea was that we are only interested in the final state

of our system, for example, that it flips from up to down.

We're not at all interested in the final state of the bath, at least for the time being.

We're not interested in that.

We have been interested in the final state of the measurement apparatus, in case this

was a measurement setup, but now we are not interested in it.

So in calculating the golden rule decay rate, really sum over all of the possible final