My question is about generalization of most simple problem in semi-classical quantum optics. Composite system of two-level systems.
single system's diagram where atomic frequency: wab=wa - wa; probing frequency: wprobe; and detuning: Delta_ab=wab - wprobe
H=h_bar/2 * Delta_ab * Sigma_z + h_bar/2 * Rabi_probe * Sigma_x = H0 + HI
Just think we have a Two-level atomic system, which is represented by 2x2 density matrix rho={{rho_aa,rho_ab},{rho_ba,rho_bb}}. As most of you know; Diagonal elements, rho_aa & rho_bb, represent populations of corresponding levels. And, non-diagonal elements, rho_ab & rho_ba, give us atomic coherence(polarization), which contains most important information about optical properties of our system like refractive index and absorption.
Practically(easier to get in numerical solvers), expectation value of pseudo-spin-x(pauli-x) operator, Expectation(Sigma_x)=Trace(rho*Sigma_x), gives us real part of coherence that corresponds to refractive index. And, expectation value of pseudo-spin-y, =Trace(rho*Sigma_y), gives us imaginary part of coherence that corresponds to absorption.
H_total=KroneckerSum(H0,H0)+KroneckerSum(HI,HI) + H_Dipole_Dipole
Long story short, where I got stuck is composite system's coherence. Density matrix of the composite system is Kronecker product of individual ones, rho_composite=tensor(rho,rho). But, what is the collective behaviour of this system when probed by light? What is the coherence(polarization) of the system? Is it the sum of the individual coherences? Or is it the product of individual coherences?
For example, if I am looking for absorption of composite system, is it expectation value of the kronecker product of two Sigma_y, tensor(Sigma_y,Sigma_y)? Or is the sum(kronecker sum), (tensor(Sigma_y,IdentityMatrix(3)) + tensor(IdentityMatrix(3), Sigma_y))?
Thanks in advance, this is my first post, I hope I haven't complicated the things.
Onur
