# Partial trace of a density matrix?

1. Apr 4, 2012

### climbon

Hi,

I'm working on a modified version of the Jayne's Cummings model and am a little confussed.

I have:
-Taken modified version of JCM Hamiltonian in Schrodinger picture.
-Used Von Neumann equation to get evolution of density matrix
-Converted to Wigner function.

I want to run numerical simulations to get time evolution of atomic inversion, mean photon number and the Phase space picture but am confused of what mathematical process I need to do in order to get these values out of my Wigner function (2x2 matrix).

I know I will need to do a partial trace, so far I'm thinking (to obtain evolution of the atomic inversion);

$$P(t)=Tr_{atom}(M \cdot \rho^{Atom}(t) ) \\ \\ Where \\ \rho^{Atom}(t) = Tr_{Field}(\rho (t)) \ and \ M = \left( \begin{array}{ccc} 1 & 0 \\ 0 & 0 \end{array} \right) \\$$

Can this be written;

$$Tr_{Atom}((\rho \cdot I)\otimes M)$$
Which is;
$$Tr_{Atom} \left( \left( \left( \begin{array}{ccc} W_{11} & W_{12} \\ W_{21} & W_{22} \end{array} \right) \cdot \left( \begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array} \right) \right) \otimes \left( \begin{array}{ccc} 1 & 0 \\ 0 & 0 \end{array} \right) \right)$$

Where $W_{nm}$ are the matrix elements of the Wigner function.

Doesn't this trace come out simply to be $W_{11} + W_{22}$?

Does this mean that by running numerical simulations and adding $W_{11} + W_{22}$ I will get the atomic inversion of the atom?