💎 Tool 13 · Cavity QED Calculator

Cavity QED Calculator

Compute the single-photon vacuum Rabi coupling g₀, cavity decay rate κ, single-atom cooperativity C, and Purcell factor for any atom–cavity system. Identify the coupling regime: strong coupling, Purcell, or weak.

In cavity QED, a single atom couples to the quantized field of an optical resonator. Three rates determine the physics: g₀ (vacuum Rabi coupling), κ (cavity field decay), and γ = Γ/2 (atomic dipole decay). Their ratio defines the cooperativity C = g₀²/(κγ) — the key figure of merit. When g₀ > κ, γ the system enters the strong-coupling regime and vacuum Rabi splitting becomes observable.

01 Cavity & Atom Parameters

Cavity Parameters

Finesse F

F = π√R/(1−R) → mirror R = —

Cavity length L (mm)

Mode waist w₀ at atom (μm)

Index of refraction n

n = 1 for free-space cavities (usual case)

Atom / Transition

Atom & transition preset

Wavelength λ (nm)

Natural linewidth Γ/2π (MHz)

Coupling fraction ξ

ξ = 1 at antinode, 0.5 for random position

02 Results

—

g₀/2π

MHz

—

κ/2π

MHz

—

γ/2π

MHz

—

Cooperativity C

dimensionless

—

Purcell Fₚ

= 2C

—

Mode vol. V

×(λ/2)³

—

FSR

GHz

—

Cav. linewidth

MHz (FWHM)

—

Sat. photons n₀

photons

—

Critical atoms N₀

atoms

$$g_0 = \sqrt{\frac{3\Gamma c^3}{2\omega^2 V}} \qquad V = \frac{\pi w_0^2 L}{4} \qquad C = \frac{g_0^2}{\kappa\gamma} \qquad \kappa = \frac{\text{FSR}}{2\mathcal{F}} \qquad \gamma = \frac{\Gamma}{2}$$

03 Coupling Rates vs Finesse

Increasing finesse reduces κ (cavity leaks more slowly) while g₀ and γ remain constant. Strong coupling is reached when the purple κ line drops below g₀. The vertical dashed line marks the minimum finesse for strong coupling with the current cavity geometry.

04 Coupling Regimes

Strong Coupling: g₀ > κ, γ — Vacuum Rabi Splitting

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.

$$\text{Strong coupling:} \quad g_0 > \kappa,\;\gamma \quad \Leftrightarrow \quad \mathcal{F} > \frac{\pi c}{4 L g_0}$$

Purcell Regime: C > 1 but g₀ < κ — Enhanced Emission

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.

Weak Coupling: C < 1 — Cavity has Little Effect

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.

🏔️

Hood Lab context: Our Li–Cs tweezer experiments don't use a cavity, but cavity QED is a natural next step — coupling a single Cs atom in a tweezer to a fiber Fabry–Pérot cavity would allow single-photon-level readout without destruction. The key challenge is that the Cs D2 line (γ/2π = 2.6 MHz) requires finesse > 50 000 for even Purcell-regime coupling in a sub-mm cavity.

📚 Key References

📄 Birnbaum et al. (2005). Photon blockade in an optical cavity with one trapped atom. Nature 436, 87–90. doi:10.1038/nature03804

📄 Thompson et al. (1992). Observation of normal-mode splitting for an atom in an optical cavity. Phys. Rev. Lett. 68, 1132. doi:10.1103/PhysRevLett.68.1132

📄 Kimble, H.J. (1998). Strong interactions of single atoms and photons in cavity QED. Phys. Scr. T76, 127. doi:10.1238/Physica.Topical.076a00127

📄 Haroche & Raimond (2006). Exploring the Quantum: Atoms, Cavities, and Photons. Oxford University Press. (Nobel Prize 2012)

📄 Reiserer & Rempe (2015). Cavity-based quantum networks with single atoms and optical photons. Rev. Mod. Phys. 87, 1379. doi:10.1103/RevModPhys.87.1379