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## Main Question or Discussion Point

**"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.