Turn geometry into a physical regime.
A cavity is not just a shiny box around an atom. Use the page in this order: compute the three rates, compare them, then decide whether the system is a spectrometer, a Purcell interface, or a strong-coupling device.
Length, waist, finesse, linewidth, and transition wavelength set g₀, κ, and γ.
Compare coherent exchange to cavity leakage and atomic free-space decay.
Strong coupling, Purcell enhancement, or weak coupling each imply different experiments.
Use reference systems and common mistakes before trusting a design.
Large dipole moment, small mode volume, and placing the atom at an antinode all increase the coherent exchange rate.
g₀ ∝ √(Γ / V)High finesse and longer photon lifetime lower κ. This makes coherent oscillation visible, but also narrows the bandwidth.
κ/2π = FSR / 2ℱγ is the unavoidable competitor. Cooperativity asks: does the cavity channel beat all the other optical modes?
C = g₀² / κγ01 Cavity & Atom Parameters
02 Results
Convention used here: rates are displayed as angular rates divided by 2π. The calculator uses a TEM₀₀ standing-wave mode volume and treats ξ as an amplitude-level overlap factor. For real experiments, multiply by Clebsch-Gordan coefficients, polarization projections, antinode position, and transverse mode overlap.
03 Coupling Rates vs Finesse
Design knobs: what should you actually change?
Most powerful lever for g₀ because g₀ ∝ 1/w₀ and C ∝ 1/w₀². The price is mirror curvature, clipping loss, alignment sensitivity, and atom-surface constraints.
Boosts g₀ through smaller mode volume and raises FSR. But short cavities are mechanically harder and can limit optical access.
Reduces κ without changing g₀. This helps strong coupling, but very high finesse narrows bandwidth and makes lock acquisition unforgiving.
Broad lines give larger g₀ but larger γ too. Narrow lines can have beautiful spectra but often require extremely small mode volume to matter.
A tiny history of making empty space non-empty
04 Coupling Regimes
When g₀ > κ and g₀ > γ, the atom–cavity system enters strong coupling. The cavity-QED Hamiltonian H = ℏg₀(a†σ⁻ + aσ⁺) drives coherent energy exchange between atom and photon faster than either can decay. The normal modes of the coupled system (the dressed states or polaritons) are split by 2g₀ — directly observable as a doublet in the cavity transmission spectrum (vacuum Rabi splitting).
Achieving strong coupling requires simultaneously large g₀ (tight mode, short cavity) and small κ (high finesse mirrors). The product gives the condition F > πFSR/(2g₀), i.e., a minimum finesse scales as cavity length.
In the bad-cavity limit (κ ≫ g₀, κ ≫ γ) the photon leaks out before a coherent Rabi oscillation completes. However, when C > 1 the cavity still dramatically enhances the spontaneous emission rate into the cavity mode. The total emission rate becomes Γ_tot = Γ(1 + 2C), and the fraction of photons emitted into the cavity mode is β = 2C/(1 + 2C). This is the Purcell effect, with Purcell factor Fₚ = 2C. Applications: efficient single-photon sources, deterministic atom–photon entanglement.
When C < 1, the cavity offers negligible enhancement of atom–photon interactions. The atom emits primarily into free-space modes. This regime is typical for initial cavity characterization experiments and for cavities used primarily for filtering or as reference resonators (e.g., PDH locking). Increasing finesse, reducing mode volume (shorter L or smaller w₀), or choosing a higher-Γ transition all increase C.
05 Reference Systems
| Group | Atom / λ | L (mm) | Finesse | w₀ (μm) | g₀/2π (MHz) | κ/2π (MHz) | γ/2π (MHz) | C | Regime |
|---|---|---|---|---|---|---|---|---|---|
| Kimble, Caltech (2005) | Cs, 852 nm | 0.042 | 460 000 | 16 | 34 | 4.1 | 2.6 | 136 | Strong |
| Vuletic, MIT (2010) | Rb-87, 780 nm | 1.7 | 130 000 | 57 | 10.4 | 0.36 | 3.0 | 32 | Strong |
| Thompson, Princeton (2013) | Sr-88, 461 nm | 1.4 | 100 000 | 100 | 3.1 | 0.27 | 16 | 2.2 | Purcell |
| Hood Lab, Purdue (ongoing) | Cs-133, 852 nm | — | — | — | — | — | 2.6 | — | — |
g₀ and κ from published papers; γ = Γ/2 from Steck data sheets. Cooperativity C = g₀²/(κγ). Use the calculator above to reproduce these numbers.
06 Common Mistakes
Three ways cavity numbers quietly go wrong
Many papers quote Γ/2π as the full natural linewidth. This calculator uses γ = Γ/2 as the dipole decay rate in C = g₀²/(κγ).
If g, κ, and γ are all consistently angular rates or all consistently divided by 2π, ratios are fine. Trouble starts when MHz and rad/s are mixed in one equation.
ξ should mean something physical: standing-wave position, polarization, Clebsch-Gordan coefficient, and transverse overlap. If ξ = 0.2, ask which imperfection caused it.