Engineering & Science
Purcell Factor (F_P) & β-Factor Spontaneous Emission Enhancement Formula
Calculate Edward Purcell’s cavity QED spontaneous emission enhancement F_P = (3 / (4 * π^2)) * (Q / V_norm) and single-photon waveguide coupling efficiency β = F_P / (F_P + 1).
Spontaneous radiative lifetime is NOT an immutable constant of nature—cavities reshape the vacuum!
In a famous one-paragraph 1946 abstract, Edward M. Purcell showed that placing a quantum emitter (quantum dot, NV center, or superconducting qubit) at the antinode of a resonant cavity with quality factor Q and normalized mode volume V_norm = V_mode / (λ / n)^3 accelerates its radiative decay rate by the Purcell factor F_P = (3 / (4 * π^2)) * (Q / V_norm) ≈ 0.075991 * (Q / V_norm)!
In a photonic-crystal nanobeam cavity with Q = 12,000 and sub-cubic-wavelength mode volume V_norm = 0.85 (λ/n)^3, F_P reaches 1,072.81—shortening a 1.0 ns InAs quantum-dot radiative lifetime below 1 picosecond and funneling β = F_P / (F_P + 1) = 99.907% of all emitted single photons into the target cavity mode!
| Nanophotonic Resonator (V_norm in (λ/n)^3) | Quality Factor Q | Purcell Enhancement F_P | Spontaneous Coupling β = F_P/(F_P+1) |
|---|---|---|---|
| Photonic-Crystal Nanocavity (V_norm = 0.85) | 12,000 | 1,072.81× | 0.99907 (99.91% Indistinguishable) |
| Micropillar Bragg Cavity (V_norm = 0.85) | 2,500 | 223.50× | 0.99555 (99.55% Mode Coupling) |
| Invalid Cavity (Q <= 0 or V_norm <= 0) | Non-physical | "Invalid Cavity Parameters" | Positivity Check |
Purcell Spontaneous Emission Enhancement Factor F_P
Evaluates (3 / (4 * PI()^2)) * (Q / V_norm) for an emitter resonant and polarization-aligned at the cavity field maximum.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Cavity Parameters", ROUND((3 / (4 * (PI() ^ 2))) * (B2 / B3), 2))Spontaneous Emission β-Factor into Cavity Mode: F_P / (F_P + 1)
Computes the fraction of radiative decays channeled into the cavity mode assuming unit free-space radiative quantum efficiency.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Cavity Parameters", ROUND(((3 / (4 * (PI() ^ 2))) * (B2 / B3)) / (((3 / (4 * (PI() ^ 2))) * (B2 / B3)) + 1), 5))Try the example
Use the original values or change them to test your understanding.