Partial trace of the density matrix

[climbon]

Hi,

I am trying to work out the atomic inversion of the Jaynes cummings model using the density matrix. At the moment i have a 2x2 matrix having used the Von neumann equation (technically in Wigner function in x and y).

Each of my matrix elements are 1st order pde's describing the evolution of the system...I want to know how do I perform a partial trace on this density matrix.

I've seen it mentioned that to work out the atomic inversion of the atom I need to do a partial trace over oscillator variables...but I don't understand how to do this mathematically...using my 4 pde's in x and y?

Any help would be great. Thanks

 

[azntoon]

.To perform a partial trace on a density matrix, you need to start by rewriting your 4 PDEs in terms of the density matrix elements. That is, express each of the 4 PDEs in terms of the elements of the density matrix (e.g. for the 2x2 matrix, this could be something like ρ11, ρ12, etc.). Once you have done that, you can then use the standard methods for performing a partial trace on a density matrix. This involves taking a trace over one of the variables (e.g. the oscillator variables) and then using the cyclic property of the trace operation to simplify the expression.