Okay, so today we are going to start our discussion of dissipation in quantum mechanics because

last time when we for example discussed the measurement of a qubit we already saw that

this is an important process where a qubit first is in the excited state and then at

some point suddenly randomly it decays. And the same happens of course with an atom that

spontaneously emits a photon at a random point in time or also the same could be said about

a cavity containing a photon where the photon then suddenly leaks out. And the question

is how to describe dissipation in quantum systems and it turns out that it's not quite

as simple as to describe dissipation in classical systems where you would just enter a friction

force in the equation of a question.

So the typical examples would be decay of an atom, a decay of a photon out of a cavity,

or more directly quantum mechanical processes such as that if you have a superposition to

start with which is coherent and has a definite phase relation then this phase might get random

and that is called dephasing.

So describing dissipation is really very important whenever you discuss time evolution in quantum

mechanics because it hardly ever happens that the time evolution is so fast that decay doesn't

take place during this time.

So our goal will be to discuss the simplest possible description of dissipation in quantum

mechanics which is known as a quantum master equation. But before we get to that I just

want to give you one example of where it is actually possible in principle to calculate

exactly what happens during such a dissipative process.

And that would be the spontaneous decay of an atom.

So what would be the proper Hamiltonian to describe this system? Well, you first include

the fact that you have your atom and again be approximated by a two-level system.

Then you have the bath of normal modes of the electromagnetic field. Think of plane

waves and vacuum. And each of those is of course represented by a harmonic oscillator.

So that would be typically called the bath similar to a heat bath known from classical

statistical physics which is the infinite reservoir which you couple to in order to

establish a equilibrium. So this can take up any amount of energy.

And then finally there will be the interaction between the atom and the bath and I want to

write in this manner sigma x would be the operator that induces the transition between

the two levels of the atom. That is essentially the dipole operator in the correct physical

setting. And f would be proportional to the electric field at the point of the atom.

And so f you could write in terms of these normal modes and it would be linear again

in these mode operators. And that is of course a Hamiltonian which in one way or another

we have seen several times.

Now it turns out that this can be solved exactly at least under the approximation of doing

the rotating wave approximation which you know just means that you keep only those terms

which approximately at least conserve the energy.

So instead of having sigma x multiplied by a plus a dagger you just keep those terms

where the atom is excited when the photon is destroyed or the atom is de-excited which

relaxes to its ground state while limiting a photon.

And then if you assume that your initial state is simply the state where the atom is excited

but everything else is in the vacuum state that means zero photons in all the modes then

an exact solution is possible. There is an exact solution of the Schrodinger equation

for this system.

And this exact solution even has a name. It is called the Weisskopf-Biegner solution.

So this dates back really to the very beginning of the new quantum mechanics around 1930.

That was one of the first applications of the new quantum mechanics in quantum field

theory.

I will not go through the solution even though it is a very beautiful piece of mathematics.