# How to model a classical bath?

1. Aug 1, 2011

### IsNoGood

Hello everybody,

I recently had to do with a model of a single Spin coupled to a very generic quantum mechanical bath (the actual point was finding pulses that manipulate the spin as accurate as possible despite having a finite length in time but as this doesn't have anything to do with my questions I'll just omit their hamiltonians):

$H = H_b + \vec{\sigma}\vec{A}$
where $H_b$ describes the internal dynamics of the bath, $\vec{\sigma}$ is the well known pauli matrix-vector and $\vec{A}$ a vector containing operators which model the interaction between the spin (in its 3 directions, of course) and the bath.
In subsequent calculations, it was important not to define either $H_b$ or $\vec{A}$ any further. To remain as generic as possible, even $[H_b,\vec{A}] \neq 0$ was assumed, only $[\vec{A},\vec{\sigma}] = 0$ was exploited later.

To put it in a nutshell:
I now made up two questions, being
1.is it possible to find a similarly generic model which describes the coupling of the spin to a classical bath and
2. what would a toy model of such a system look like?

Every hint is appreciated.

2. Aug 1, 2011

### SpectraCat

If the components of the classical bath have magnetic moments associated with them, then you will get a *direct* coupling to the classical bath, otherwise there should be no *direct* coupling as far as I can tell. Of course, if the spins are carried by particles, and the classical bath is represented as an ensemble of particles, then there will be an indirect coupling through collisions between the particles. Another possibility is that if there is an external magnetic field present, its effects on the collisional dynamics between the particles carrying spins and the bath particles would need to be accounted for.

Does that help?

3. Aug 8, 2011

### IsNoGood

I guess what I was searching for can be best described by
$\vec{\sigma}\vec{A} \rightarrow \vec{\sigma}\vec{c}(t)$
where $\vec{c}(t)$ is a vector of gaussian distributed random variables modeling the coupling of the spin to the classical bath in every direction. (By the way, does anybody have an idea what could reasonably be chosen to be the mean $\mu$ of this gaussian distribution?)

However, there's still one question left:
In the case of a classical bath, is there anything corresponding to the inner dynamics $H_b$ of the quantum bath?

As always, every hint is greatly appreciated.