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Atomic Coherence(Polarization) of Two Dipole-Dipole Coupled Atoms

  1. Aug 22, 2012 #1
    My question is about generalization of most simple problem in semi-classical quantum optics. Composite system of two-level systems.

    single system's diagram: http://ars.sciencedirect.com/content/image/1-s2.0-S0030401807009777-gr1.jpg [Broken]

    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.

    composite system's diagram: http://i.imgur.com/lffy6.png

    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

    ps1: I used kronecker sum for abbreviation,

    KroneckerSum(A,B)=TensorProduct(A,IdentityMatrix(dimB))+TensorProduct(IdentityMatrix(dimA),B)

    ps2: maybe I am misleading and creating a bias in your answers. Getting a spectrum by fourier transform makes much more sense now, because states of the atoms will be entangled during non-unitary evolution and will even be mixed. However, I can not relate correlation functions in full-quantum statistical theory and coherences of semi-classical one
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Aug 22, 2012 #2
    Or, would it make more sense to think about coherence between two symmetric-asymmetric (Dicke basis or so called singlet-triplet states of composite system) states?

    Even when we start in a density matrix that is product of individual ones, (non)unitary evolution may complicate things by making them entangled may be?
     
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