Now we want to continue our discussion of dissipative quantum systems.

Last time we introduced the simplest general description of dissipative quantum systems

and that is Limb-Lod Markov master equations.

So the idea is that instead of having an evolution equation for the wave function or for the state,

we will have an evolution equation for the density matrix and that contains a term

that describes coherent evolution according to some Hamiltonian,

that is the simple von Neumann evolution equation,

plus other terms that describe various dissipation processes.

And these can be written as a sum over all of these dissipation processes

with some super operators that have to be specified acting on the density matrix.

Each of these Limb-Lod super operators actually can be expressed in terms of some relaxation operator

that belongs to a specific kind of dissipation channel and has to be multiplied with the corresponding weight.

Now these relaxation operators act on the density matrix in the following way.

The principle of R acting on the density matrix is quite simple.

First you want to act on the density matrix both from the left and from the right.

If you remember the fact that the density matrix for a pure state would be just a projectile onto the pure state.

So if you want to act on the pure state you have to act both from the left and from the right.

That would be R from the left and R dagger from the right.

Now this would actually describe the dissipation process itself.

But in order to have conservation of probability if you have a process that gives you probability ending up in a new state

you also have to eliminate the corresponding probability in the state where you came from.

And this is taken care of by the following two terms.

Okay, so this is the full action of the Lindblad superoperator on the density matrix.

And now we just practiced how this can be applied to the two-level system

and we ended by starting with a description of this Lindblad master equation in terms of the damped harmonic oscillator.

So for the damped harmonic oscillator the idea is that if you have some damping

which in the classical equations would be described by a simple velocity dependent damping term

then quantum mechanically you can describe things in the regime of weak coupling.

You can describe things by such a Lindblad master equation

and the relaxation operator you need for this purpose is just the annihilation operator.

This somehow makes sense. You annihilate a quantum of energy in the harmonic oscillator and that gives rise to the dissipation.

Now we then asked how does the evolution equation for the density matrix look like.

In particular how does for example the photon number decay as a function of time

and we found that the equation for this is very simple.

Mainly the expectation value of the photon number decays exponentially with the rate given by gamma.

And now all of this is true for zero temperature.

We'll turn to the finer temperature case in a minute.

But first let's ask what other interesting observables are there whose decay you might want to describe.

Photon number that is the energy is one of them

but another example is certainly the position or the momentum

and quantum mechanically it's even more convenient to describe the decay of the variable that contains both of them

both position and momentum and you know that is the annihilation operator A.

So the question is how does the expectation value of A decay.

And you know from our discussion of coherent states that if you displace the ground state and A acquires a non-zero expectation value

that would be a complex number. The real part would be connected to the expectation value of the position.

The imaginary part would be connected to the expectation value of momentum.

So how does this amplitude this complex amplitude of motion decay in the course of time.

Well if you plug in again the master equation similar to what we discussed for the photon number last time

then in the end you arrive at a very simple equation of motion namely the amplitude again decays exponentially

but now with a rate gamma half.