(9820) Two-level-atom emission - radiance and superradiance

A toy model for atom is a two level system (TLS). It can be visualized as a mirror-symmetric two site system. The sites have separation \( a_0 \), and the hopping amplitude between them is \( \omega/2 \). The interaction between the EM field and the atom in the dipole approximation is \( V=-(e/c) v \cdot A(0) \).

(1) Write what are the matrix elements \( D_j \) of the dipole operator \( \hat{r} = (\hat{x},\hat{y},\hat{z}) \).

(2) Express the matrix elements of the velocity operator \( \hat{v} \) using those of \( \hat{r} \). Exploit the identity \( v=i[H,r] \).

(3) Given \( \vec{D} \), find expression for the matrix element that allows spontaneous emission from an excited atom.

(4) Use Fermi golden rule to find the decay rate \( \Gamma_0 \). Sum over all possible directions and polarization.

(5) Consider system of \( N \) closely packed TLS atoms. Formally each TLS can be regarded as spin 1/2. Explain why the total angular momentum \(J^2 \) is constant of motion. How many multiplets are there with \( j=N/2 \), with \(j=N/2-1\), and with \( j=N/2-3 \).

(6) Assume that at \( t=0 \) all the atoms are in the excited state. Prove that the initial rate of radiance is \( \Gamma = N\Gamma_0 \), while after a transient one obtains the maximum rate \( \Gamma \approx (N^2/4)\Gamma_0 \), aka superradience.

Tip: The solution for \( \Gamma_0 \) can be found at the end of Section [50] of the lecture notes.