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Partial trace of a density matrix?

  1. Apr 4, 2012 #1
    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);

    [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?
     
  2. jcsd
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