Vacuum Rabi Oscillations: Time-Domain Cavity QED (2D)

Atom-Cavity System

WEAK COUPLING

Coupling strength g 0.30

Cavity loss κ 1.00

Atom decay γ 0.20

Atom-cavity detuning Δ 0.00

Live Readout (RK4, from the trace)

Measured period

—

Theoretical Ω

—

Swaps before decay

—

Cooperativity C

—

Max cavity |a|²

—

Energy remaining

—

How it works

The frequency-domain 3D version reads the strong-coupling threshold off a steady-state transmission formula. This companion instead integrates the actual time-domain coupled-amplitude equations — the semiclassical normal-mode-splitting model, the same linear system that underlies the quantum Jaynes-Cummings result at the single-excitation level — with 4th-order Runge-Kutta:

da/dt = −(κ/2)·a − i·g·σ
dσ/dt = −(γ/2 + iΔ)·σ − i·g·a
a(0)=0, σ(0)=1   (atom excited, cavity empty — "trigger excitation")

Below the strong-coupling threshold the excitation leaks out through κ or γ before it can return — |σ(t)|² and |a(t)|² just decay, no oscillation. Above threshold (g > |κ−γ|/4, on resonance) the excitation swaps coherently between atom and cavity — the vacuum Rabi oscillation — before the envelope eventually decays. The "Measured period" is read directly off the peaks of the integrated |σ(t)|² trace and checked against the closed-form Ω = 2√(g² − ((κ−γ)/4)²), so agreement between the two is a genuine numerical cross-check, not a repeated formula.

Atom excitation |σ(t)|² Cavity photon |a(t)|²