A two-level atom coupled to a single cavity mode — vacuum Rabi oscillations, collapse and revival
Jaynes-Cummings Hamiltonian
H = ωa†a + ω₀σz/2 + g(aσ₊ + a†σ₋)
In the rotating frame with detuning Δ = ω − ω₀:
H_JC = ℏΔa†a + ℏg(aσ₊ + a†σ₋)
The dressed states (polaritons) at photon number n:
Ω_n = g√(n+1) (Rabi frequency)
Starting in |e,n⟩ (excited atom, n photons):
⟨σz⟩(t) = −cos(2Ω_n t)
For a coherent state |α⟩ with ⟨n⟩=|α|², the Rabi oscillations collapse due to dephasing between Fock components, then revive at T_rev = 2π√⟨n⟩/g.
Collapse, Revival & Cat States
A coherent state is a superposition of Fock states:
|α⟩ = e^{-|α|²/2} Σ_n (α^n/√n!) |n⟩
Each Fock component oscillates at Ω_n = g√(n+1), causing the inversion to collapse as they dephase, then periodically revive when they re-phase.
At half-revival time, the cavity field is in a Schrödinger cat state — a superposition of two coherent states with opposite phases. The Wigner function W(α) of this state shows negative values, a hallmark of quantum non-classicality.
T_collapse ~ 1/(g√⟨n⟩)
T_revival ~ 2π√⟨n⟩/g