Partial trace of a density matrix?

[climbon]

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);

$$P(t)=Tr_{atom}(M \cdot \rho^{Atom}(t) ) \\ \\ Where \\ \rho^{Atom}(t) = Tr_{Field}(\rho (t)) \ and \ M = \left( \begin{array}{ccc} 1 & 0 \\ 0 & 0 \end{array} \right) \\$$

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 $W_{nm}$ are the matrix elements of the Wigner function.

Doesn't this trace come out simply to be $W_{11} + W_{22}$?

Does this mean that by running numerical simulations and adding $W_{11} + W_{22}$ I will get the atomic inversion of the atom?

 

[eljose79]

Hello,

Thank you for sharing your progress on your modified version of the Jayne's Cummings model. It sounds like you have made good progress so far. To answer your question, yes, your proposed method for obtaining the atomic inversion using the Wigner function is correct. The trace operation you have described will indeed give you the atomic inversion of the system, which is the expectation value of the Pauli operator I (also known as the identity operator). This is because the trace operation essentially sums up the diagonal elements of the matrix, which in this case correspond to the expectation values of the Pauli operators.

However, I would like to point out that in order to get the atomic inversion, you will need to perform the trace operation on the density matrix of the atom, not the field. This is because the atomic inversion is a property of the atom, not the field. So your equation should be P(t)=Tr_{atom}((\rho^{Atom} \cdot I)\otimes M).

As for obtaining the mean photon number and the phase space picture, you will need to perform a similar trace operation, but on different operators. For the mean photon number, you will need to use the number operator (a^{\dagger}a), and for the phase space picture, you will need to use the position and momentum operators (x and p). The trace operation in these cases will give you the expectation values of these operators, which will correspond to the mean photon number and the phase space picture, respectively.

Overall, your approach is correct and you can use numerical simulations to obtain these values by adding up the diagonal elements of the Wigner function. I hope this helps clarify your confusion. Good luck with your simulations!