User:Jambaugh/Sandbox/Decoherence

Some Tools
In order to describe meaningfully the interplay of system and environment we are going to need a quantum representation of a simplified environment.

Let us begin with a simple spin network. $$\psi_k =\left(\begin{array}{c}\psi^\uparrow_k \\ \psi^\downarrow_k\end{array}\right)$$ with $$k = 1,2,\cdots N$$ presuming $$N$$ a sufficiently large number. Now we also consider an interaction Hamiltonian which "prefers" correlation of z-components of spins:

$$ H_{int}=-\epsilon\sum_{k=1}^{N-1}\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right)_k\otimes\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right)_{k+1}$$

so that...

$$H_{(k),(k+1)} = -\epsilon \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\end{array}\right)$$

TODO: Treat these as pointers. Rotate in proportion to the z-spin but only very very weakly so that

A Simple Example
Consider a two component system initially independently sharp: $$\Psi = \psi_1\otimes\psi_2$$, allow that $$\psi_1 =\left(\begin{array}{c}1 \\ 0\end{array}\right)$$