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

In summary, the conversation discusses the generalization of a simple problem in semi-classical quantum optics, specifically in a composite system of two-level systems. The density matrix of the individual systems is represented by diagonal and non-diagonal elements, which correspond to the populations and coherence of the system, respectively. The expectation values of pseudo-spin operators can give information about refractive index and absorption. The discussion then moves on to the composite system and the behavior of its coherence when probed by light. There is a question about whether the coherence of the composite system is the sum or product of the individual coherences. The use of Kronecker sum and tensor product is explained. The concept of entanglement and its potential influence on coherence is also
  • #1
necrite
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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
 
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  • #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?
 

1. What is Atomic Coherence (Polarization)?

Atomic coherence, also known as polarization, refers to the alignment of the dipole moments of atoms in a specific direction. It occurs when two dipole-dipole coupled atoms interact with each other and become synchronized in their oscillations.

2. How do Two Dipole-Dipole Coupled Atoms affect each other's polarization?

When two dipole-dipole coupled atoms interact, their dipole moments align in the same direction, leading to synchronized oscillations. This results in an increase in the polarization of both atoms, making them more coherent.

3. What factors influence the Atomic Coherence of Two Dipole-Dipole Coupled Atoms?

The atomic coherence of two dipole-dipole coupled atoms can be influenced by various factors such as the strength of the dipole-dipole coupling, the distance between the atoms, and the frequency of the applied electromagnetic field.

4. Why is Atomic Coherence important in quantum information processing?

Atomic coherence is crucial in quantum information processing as it allows for the creation of entangled states between two atoms, which is essential for various quantum computing algorithms. It also enables the generation of quantum entanglement and quantum teleportation, which are essential for quantum communication and cryptography.

5. How is Atomic Coherence (Polarization) measured?

Atomic coherence can be measured through techniques such as Rabi oscillation measurements, Ramsey spectroscopy, and fluorescence detection. These methods involve exciting the atoms with a laser and detecting the resulting fluorescence or changes in the atomic energy levels to determine the degree of atomic coherence.

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