Dicke Superradiance

Dicke Superradiance

When $N$ inverted emitters are confined within a wavelength of one another, their dipoles couple through the shared radiation field. The collective emission rate scales as $N^2$, not $N$; the radiated pulse is shorter by a factor of $N$; and the angular pattern beams along the major axis of the sample rather than being isotropic. This is Dicke superradiance, after R. H. Dicke's 1954 paper Coherence in Spontaneous Radiation Processes.

The Cooperation Length

Dicke distinguished two limits by the dimensionless cooperation number — the number of inverted atoms within $(\lambda/4)^3$:

$$N_c = n \left(\frac{\lambda}{4}\right)^3$$

When $N_c \lesssim 1$, atoms radiate independently; total intensity is linear in $N$. When $N_c \gtrsim 1$, the system's radiation reaction couples atoms phase-coherently and the emission becomes a single quantum state — the superradiant state — whose decay is sped up by a factor of $N$.

This is exactly the threshold $N\lambda^3 / 64 > 1$ that governs cooperative emission, up to the geometric prefactor: $64 = 4^3 \cdot \pi^?$ depending on whether you count the cube, the sphere, or the spherical-shell version. The threshold is one physics, with different names in different communities.

Why N² Not N

A heuristic: in the independent-emitter picture, the field at the detector is the incoherent sum of $N$ amplitudes — intensity $\propto N |A|^2$. In the cooperative picture, the dipoles share a phase, so the field is the coherent sum — intensity $\propto |NA|^2 = N^2 |A|^2$.

The mass-action accounting:

| regime | dipole phasing | intensity scaling | pulse duration | |---|---|---|---| | independent | random | $N$ | $\Gamma^{-1}$ | | superradiant | aligned | $N^2$ | $(N\Gamma)^{-1}$ |

The integrated energy $N^2 \cdot (N\Gamma)^{-1} = N \cdot \Gamma^{-1}$ is the same in both regimes — energy conservation is not violated, temporal concentration is.

Where it Lives in Real Systems

At Na D₂ ($\lambda = 589$ nm), $\lambda/4 = 147$ nm. The default operating density in industrial sodium-vapor lamps is $\sim 10^{21}$ cm⁻³ — so $N_c \sim 10^{21} \cdot (1.47 \times 10^{-5})^3 \approx 3 \times 10^6$, deep into the superradiance basin.

Stated bluntly: every sodium lamp ever made operates in the superradiant regime, and no industrial diagnostic has measured this. The standard radiation-trapping analysis (see 02-holstein-radiation-trapping) predicts dim, deeply-trapped emission. The actual emission is bright, fast, and partially coherent — but you only see the difference if you resolve nanosecond timescales and angular distributions, which the lamp industry never did.

The Intensity Ratio

From plasma-solvers::dicke_intensity_ratio(N, eta):

$$\frac{I_{\text{coop}}}{I_{\text{indep}}} = 1 + (N - 1)\eta$$

where $\eta \in [0, 1]$ is the cooperation factor — the product of geometric overlap (how aligned the dipoles are) and spectral overlap (how phase-coherent the line is over the sample lifetime).

• $\eta = 0$: independent emitters, ratio = 1.
• $\eta = 1$: perfect Dicke superradiance, ratio = $N$.

Real systems land somewhere in between. The Na D₂ line at saturated vapor density has $\eta \sim 0.1$–$0.3$ depending on Doppler vs collisional broadening — already enough for order-of-magnitude intensity enhancements.

## See also - 01-nlambda3-cooperative-threshold — the threshold this regime sits above - 02-holstein-radiation-trapping — what Dicke replaces - R. H. Dicke, Coherence in Spontaneous Radiation Processes, Phys. Rev. 93, 99 (1954) - M. Gross & S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Phys. Rep. 93, 301 (1982)