- #1
climbon
- 18
- 0
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?
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?