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Straightforward computation

  1. Mar 25, 2007 #1
    "straightforward computation..."

    Hello. I am reading a paper that discusses moving from a regular [tex] NxN [/tex] dimensional Hilbert space into and [tex]N^2 X N^2[/tex] dimensional Liouville space. The density matrix can be re-written as a [tex]1 X N^2 [/tex]vector by stacking the rows and the Liouvile equation can be re-written as [tex]N^2 X N^2 [/tex]"super-operators" acting on the new density-matrix "vector". The actual form of this super-operator is given by "a straightforward computation" where the field free and interaction Hamiltonians are given by
    [tex] H_o=\left(\begin{array}{cc}
    -hw & 0 \\
    0 & hw
    \end{array}\right) [/tex]
    and
    [tex] H_1=\left(\begin{array}{cc}
    0 & d1 \\
    d1 & 0
    \end{array}\right) [/tex]

    The Liouville space super operators describing these in the expanded space are:
    [tex] L_o=
    \left(\begin{array}{cccc}
    0 & 0 & 0 & 0 \\
    0 & -hw &0 & 0\\
    0 & 0 & hw & 0\\
    0 & 0 & 0 & 0
    \end{array}\right) [/tex]
    and

    [tex] L_1=
    \left(\begin{array}{cccc}
    0 & -d1 & d1 & 0 \\
    -d1 & 0 & 0 & d1\\
    d1 & 0 & 0 & -d1\\
    0 & d1 & -d1 & 0
    \end{array}\right) [/tex]

    It looks like they're takind a direct product with something, but I can't tell with what or why. They make is sound like it's all just a pretty standard way to write a matrix in a larger space, so if anyone recognizes what's going on here, I'd really happy for your help. Thanks in advance.
     
  2. jcsd
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