User:Jambaugh/Sandbox/Decoherence

Quantum 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. $\psi _{k}=\left({\begin{array}{c}\psi _{k}^{\uparrow }\\\psi _{k}^{\downarrow }\end{array}}\right)$ 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 $\psi _{1}=\left({\begin{array}{c}1\\0\end{array}}\right)$