One atom in one mode: the Jaynes-Cummings model

SI units throughout; the course's metric signature is (+, −, −, −), though no four-vector appears here. Quoted from topic 11.08: the leading atom-field term changes the photon number by one, and in the dipole approximation its resonant matrix elements equal those of −d⋅E (Quantum Mechanics II). Topic 11.03's construction, applied to one standing-wave mode of a cavity, gives E(r) = E_vac f(r) ε̂ (a + a†), with f real, |f| ≤ 1, mode volume V = ∫f² d³r, and E_vac = √(ħω/(2ε₀V)) the vacuum field of topic 11.05 (quoted). Keep two atomic levels; parity kills the diagonal elements, so d⋅ε̂ = d(σ₊ + σ₋) with d = ⟨e|d⋅ε̂|g⟩ made real by a choice of phase. Then −d⋅E = −ħg f(r₀)(σ₊ + σ₋)(a + a†) with ħg = dE_vac at an antinode: derived, and the sign is removed by relabelling |e⟩ → −|e⟩. Dimensions: C m × V m⁻¹ = J. An atom with d = ea₀ = 8.48 × 10⁻³⁰ C m in a mode of wavelength 1.00 µm and volume λ³ feels E_vac = 1.06 × 10⁵ V m⁻¹ and has g/2π = 1.36 GHz; a mode a thousand times larger gives 42.9 MHz.

Multiply out (σ₊ + σ₋)(a + a†). In the interaction picture σ₊a carries e^{(i(ωₐ − ω_c)t)}, slow near resonance, and σ₋a† its conjugate, while σ₊a† and σ₋a carry e^{(±i(ωₐ + ω_c)t)} and turn through a cycle every π/ω. Over times of order 1/g they average away when g ≪ ωₐ + ω_c, and second-order perturbation theory puts their leftover effect at a shift of order g²/(ωₐ + ω_c), the Bloch-Siegert shift. Dropping them is the rotating-wave approximation. Now [a†a, aσ₊] = −aσ₊ and [σ₊σ₋, aσ₊] = +aσ₊, so N = a†a + σ₊σ₋ commutes with aσ₊, with a†σ₋ and with both free terms: derived. The dropped pair changes N by ±2, which is exactly why it had to go. For the circuit of Wallraff et al. (2004), g/2π = 5.8 MHz on a 6.04 GHz resonator (quoted): g/ω = 9.6 × 10⁻⁴, and the neglected shift, about g²/(2ω), is 2π × 2.8 kHz, 2.4 × 10⁻⁴ of the splitting 2g. Circuits built to reach g/ω above 0.1 (quoted) break the approximation; then only the parity (−1)^N survives and the blocks are no longer 2 × 2.

For N = n + 1 the block holds |e, n⟩, energy ħ(nω_c + ωₐ), and |g, n + 1⟩, energy ħ(n + 1)ω_c, joined by ħg√(n + 1) from a†|n⟩ = √(n + 1)|n + 1⟩. A 2 × 2 Hermitian matrix with diagonal difference ħδ, δ = ωₐ − ω_c, and off-diagonal ħg√(n + 1) has eigenvalues at its mean ± (ħ/2)√(δ² + 4g²(n + 1)): derived. On resonance the dressed states are (|e, n⟩ ± |g, n + 1⟩)/√2 and the splittings run 2g, 2.83g, 3.46g, 4g; the lone ground state |g, 0⟩ is unshifted. Three limits check it. As g → 0 the bare levels return. For |δ| ≫ g√(n + 1) the shifts become ±ħg²(n + 1)/δ, Quantum Mechanics II's second-order perturbation theory: at δ = 10g the exact n = 0 shift is 0.0990g against 0.1g. For n ≫ 1 the splitting tends to 2g√n, which is dE₀/ħ for a classical field of amplitude E₀ = 2E_vac√n, the semiclassical Rabi frequency; at n = 100 they differ by 0.5%. Two coupled harmonic oscillators, which is what a classical Lorentz atom in a cavity is, give an evenly spaced ladder, so the √2 step from the first doublet to the second is a quantum mark, resolved in a circuit by Fink et al. (2008) (quoted).

Start the atom in |e⟩ with the mode empty. |e, 0⟩ lies in the N = 1 block, so on resonance |ψ(t)⟩ = e^(−iωt)[cos(gt)|e, 0⟩ − i sin(gt)|g, 1⟩] and Pₑ = cos²(gt) = (1 + cos 2gt)/2: derived. By t = π/(2g) the photon is in the mode with certainty and by π/g it is back in the atom; at gt = π/4 atom and mode are maximally entangled and the atom alone is in the mixed state ½(|e⟩⟨e| + |g⟩⟨g|). Nothing decays because the block has only two energies: every amplitude is a sum of two phases e^(−iEt/ħ), and so periodic. A semiclassical atom cannot do this at all: in a classical field of zero amplitude the coupling vanishes, and an atom in |e⟩, with ⟨d⟩ = 0, radiates no field of its own, so it stays excited for ever. In the spectrum the same physics is two lines 2g apart, the vacuum Rabi splitting, resolved once g exceeds the cavity and atomic linewidths. Measured (quoted): Brune et al. (1996) saw circular Rydberg atoms oscillate in an empty 51 GHz cavity at 2g/2π ≈ 47 kHz, and Wallraff et al. (2004) resolved 2g/2π = 11.6 MHz in a circuit, a vacuum Rabi period of 86 ns.

Let the mode start in a coherent state with n̄ = |α|², whose photon numbers are Poisson, Pₙ = e^(−n̄)n̄ⁿ/n!, with spread Δn = √n̄ (topic 11.07, quoted). Each component |e, n⟩ Rabi-oscillates in its own block, so the inversion is W = Pₑ − P_g = Σₙ Pₙ cos(2g√(n + 1)t), exactly. For n̄ ≫ 1 expand the frequency about n̄: 2g√(n + 1) ≈ 2g√(n̄ + 1) + g(n − n̄)/√(n̄ + 1). Averaging over a near-Gaussian spread of variance n̄ multiplies the cosine by e^{(−g²t²n̄/(2(n̄ + 1)))}, so W ≈ cos(2g√(n̄ + 1)t) e^{(−g²t²/2)}: derived. The collapse time t_c = √2/g does not depend on n̄, because the spread of Rabi frequencies, Δn times their slope g/√(n̄ + 1), stays close to g. For n̄ = 10, P₉ = P₁₀ = 0.125, the mean Rabi frequency is 6.63g, and the exact sum falls to 1/e at gt = 1.47, so only about one and a half Rabi cycles are seen. The collapse is dephasing: nothing has left the atom and the mode. Nor does it prove quantisation. A classical field of fixed amplitude 2E_vac√n̄ would Rabi-oscillate at 2g√n̄ for ever, but one whose intensity is spread as widely would collapse in the same way.

The exact sum is over integers, and neighbouring terms differ in frequency by 2g(√(n + 2) − √(n + 1)) ≈ g/√(n̄ + 1). They come back into phase when that difference times t reaches 2π: tᵣ ≈ 2π√(n̄ + 1)/g ≈ 2π√n̄/g, derived. For n̄ = 10 the formula gives gt = 20.8 and the exact sum peaks at 20.9 with 0.57 of full contrast; later revivals near ktᵣ come lower and broader, because √n is curved, and in the end overlap. Collapse and revival separate cleanly since tᵣ/t_c ≈ π√(2n̄), 14 at n̄ = 10. No classical field does this: a fixed amplitude gives no collapse, and a continuously spread one gives Rabi frequencies that fill a band and never rephase. The revival is the photon's discreteness read off the atom alone. Measured (quoted): Rempe, Walther and Klein (1987) sent rubidium Rydberg atoms through a superconducting cavity on a 21.5 GHz transition, chose their speed to set the interaction time, and saw the excitation collapse and revive, in a field built up by earlier atoms rather than a coherent one. It needs both lifetimes long compared with tᵣ, and a single mode.