Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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);

[tex]

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)

\\

[/tex]

Can this be written;

[tex]

Tr_{Atom}((\rho \cdot I)\otimes M)

[/tex]

Which is;

[tex]

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)

[/tex]

Where [itex]W_{nm}[/itex] are the matrix elements of the Wigner function.

Doesn't this trace come out simply to be [itex]W_{11} + W_{22}[/itex]?

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

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# Partial trace of a density matrix?

Can you offer guidance or do you also need help?

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