Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Partial trace of a density matrix?

  1. Apr 4, 2012 #1

    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 [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?
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Threads - Partial trace density Date
Partial trace of the density matrix Oct 15, 2012
Partial trace of density matrix Aug 2, 2012
Partial Trace Question Nov 16, 2011
Density Operators, Trace and Partial Trace Sep 6, 2010
Second quantization and partial traces Apr 27, 2009