Dicke Symmetric Emission Ladder

Worked Example

Let us calculate the emission rates for an ensemble of 4 atoms (N = 4) starting with 2 excitations (n = 2) and a single-atom decay rate of Γ = 1. The collective spontaneous emission rate from a fully symmetric state is given by the formula:

\Gamma_(coll) = Γ · n · (N - n + 1)

1. Identify the parameters: N = 4, n = 2, Γ = 1.
2. Calculate the independent rate if the atoms did not interact: \Gamma_(ind) = Γ · n = 1 · 2 = 2.
3. Calculate the collective rate: \Gamma_(coll) = 1 · 2 · (4 - 2 + 1) = 1 · 2 · 3 = 6.
4. Calculate the peak ladder rate which occurs at the maximum possible collective emission for this system: ⌊ (N+1)² / 4 ⌋ Γ = ⌊ 25 / 4 ⌋ · 1 = 6.

Notice how the collective rate (6) is significantly larger than the independent rate (2). This enhancement is due to the constructive interference of the emission pathways for identical atoms in the symmetric superposition state.

Experiment

You can interact with the module to see how the collective emission rate evolves as the number of excitations changes.

1. Open the interactive module at https://riano.app/module/dicke-symmetric-emission-ladder.
2. Set Atoms to 8.
3. Set Excitations to 4.
4. Set Single-atom rate to 1.
5. Observe the Collective rate value, which displays 20. Compare this to the Independent rate of 4.
6. Change Excitations to 8 (fully inverted). Observe that the initial collective rate is 8, exactly equal to the independent rate. The enhanced peak rate of 20 will only be reached later as the system cascades down the ladder to n = 4.

Limitations

This idealized model assumes all N atoms are identical two-level systems confined within a subwavelength volume such that dipole-dipole interactions perfectly couple them into a fully symmetric manifold (a state with total angular momentum J = N/2). It neglects dephasing, spatial disorder, non-symmetric states, and dipole-dipole frequency shifts. Furthermore, the equations describe the instantaneous emission rate for a specific excitation count n; the real physical system cascades dynamically down the ladder of states. An invalid input where n > N is unphysical and results in an undefined state.

References

The concepts demonstrated in this module are based on the original formulation of superradiance and modern reviews of collective emission from symmetric states.

• Introduction to superradiance (arXiv) - A comprehensive review on collective spontaneous emission and its enhanced decay rates.
• PIQS: Permutational Invariant Quantum Solver - Documentation demonstrating numerical simulations of superradiance and Dicke states, supporting the exact rate equations used in this model.