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Jaynes-Cummings with losses
A two-level atom on resonance with one cavity mode, in the one-excitation subspace, has amplitudes a (atom excited, cavity empty) and b (atom in ground state, one photon): da/dt = -(γ/2) a - i g b, db/dt = -(κ/2) b - i g a. Without loss, a = cos(g t) and the populations swap with the vacuum Rabi frequency ΩR = 2g.
Strong and weak coupling
The amplitudes decay with eigenvalues −(κ+γ)/4 ± √(((κ−γ)/4)2 − g2). If g exceeds |κ−γ|/4 the system oscillates (strong coupling, the spectrum splits into two peaks 2√(g2 − ((κ−γ)/4)2) apart); otherwise it is overdamped. The cooperativity C = g2/(κγ) measures how much coupling beats loss.
Rates are in units of 106 s-1 and time in microseconds. The animation runs slower than real time so the swaps are visible.
Honest simplifications: a single excitation only (no photon-number ladder or quantum jumps), resonant atom and cavity, one atom at a field antinode, and no dephasing.
What to try
- Use Ideal (lossless) and read the swap period pi / g from the first graph.
- Use Weak coupling: the oscillation vanishes and the spectrum has one peak.
- Raise kappa from 1 to 10 MHz with g = 6: the two spectral peaks merge.
- Watch the second graph: the lost probability grows to 1 as the quantum leaks out.