Jaynes-Cummings Model: One Atom Trading a Photon with a Cavity

The Physical Setup: One Atom, One Mode

The Jaynes-Cummings model strips light-matter interaction down to its irreducible core. Take a two-level atom — a ground state |g⟩ and an excited state |e⟩ separated by energy ħω_a — and place it inside a resonator (a cavity, or a superconducting LC circuit) that supports a single electromagnetic mode of frequency ω_c. Every other atomic level and every other field mode is ignored.

The atom is treated as a spin-½: the Pauli operator σ_z measures whether it is excited, while σ₊ = |e⟩⟨g| and σ₋ = |g⟩⟨e| flip it up and down. The field mode is a quantum harmonic oscillator with creation and annihilation operators a† and a, so a†a counts photons. Crucially, both subsystems are quantized — this is what separates the JCM from the semiclassical Rabi problem, where the field is a fixed classical wave.

• Atom: energies ±½ħω_a, transition ħω_a (often optical, ~1–2 eV, or microwave, ~µeV).
• Field: ladder of Fock states |n⟩ with energies ħω_c(n + ½).
• Coupling: the dipole of the atom feels the electric field of the mode, with strength g.

The Hamiltonian and the Rotating-Wave Approximation

The full dipole coupling between atom and field contains four processes. Writing g for the coupling rate, the interaction is proportional to (a† + a)(σ₊ + σ₋). Two of these terms conserve energy near resonance — a†σ₋ (atom drops, photon created) and aσ₊ (photon absorbed, atom excited). The other two, a†σ₊ and aσ₋, would simultaneously excite both or de-excite both, and they oscillate rapidly at frequency ω_a + ω_c.

The rotating-wave approximation (RWA) discards those fast counter-rotating terms, valid when the coupling is small compared to the frequencies (g ≪ ω_c, ω_a) and the detuning Δ = ω_a − ω_c is small. What remains is the Jaynes-Cummings Hamiltonian:

H = ħω_c a†a + ½ħω_a σ_z + ħg(a†σ₋ + aσ₊)

The payoff is enormous: the RWA gives H a conserved excitation number N = a†a + σ₊σ₋. Because N commutes with H, the infinite-dimensional problem block-diagonalizes into an endless stack of independent 2×2 blocks — one for each pair {|e,n⟩, |g,n+1⟩}. Each block is trivially diagonalized, which is why the JCM is one of the very few exactly solvable models of interacting quantum fields.

Dressed States, the JC Ladder, and Vacuum Rabi Splitting

Within each excitation manifold the atom-field states hybridize into dressed states |n,±⟩ — superpositions of |e,n⟩ and |g,n+1⟩. On resonance (Δ = 0) the two mix equally, and the energy eigenvalues are:

E(n,±) = ħω_c(n + 1) ± ħg√(n + 1)

The dressed pair is split by 2g√(n+1). This √(n+1) scaling is the fingerprint of the quantized field: the rungs of the Jaynes-Cummings ladder are unevenly spaced, growing as √1, √2, √3… A classical field would give evenly spaced, equally split levels.

• With one photon (n = 0), the splitting is 2g — the celebrated vacuum Rabi splitting, present even for the lowest excitation.
• Off resonance (Δ ≠ 0) the general splitting is ħ√(Δ² + 4g²(n+1)), so the two dressed levels never actually cross — an avoided crossing.

Because the ladder is anharmonic, driving |g,0⟩ → |1,±⟩ does not resonantly reach the next rung: the atom-cavity system behaves as an artificial atom with photon-number-dependent transitions, the basis for single-photon nonlinear optics.

Characteristic Numbers and a Worked Example

The dynamics on resonance are cleanest for a definite photon number. If the field starts in Fock state |n⟩ and the atom in |e⟩, the excitation oscillates coherently at the quantum Rabi frequency Ω_n = 2g√(n+1). The probability of finding the atom still excited is:

P_e(t) = cos²(g√(n+1) · t)

Even with zero photons (n = 0), an excited atom oscillates at Ω₀ = 2g — the vacuum Rabi oscillation, driven purely by the electromagnetic vacuum.

• Rydberg microwave cavity QED (Haroche-type): g/2π ≈ 25–50 kHz, cavity ω_c/2π ≈ 51 GHz; a vacuum Rabi half-cycle takes ~10 µs, feasible in ultrahigh-Q niobium cavities (photon lifetime up to ~0.1 s).
• Optical cavity QED (Kimble-type): g/2π ≈ 10–100 MHz with cesium atoms; strong coupling requires g > κ, γ (cavity and atomic decay rates).
• Superconducting circuit QED: g/2π ≈ 10–300 MHz at ω_c/2π ≈ 6 GHz — roughly 0.1–5% of the transition frequency, giving Rabi periods of nanoseconds.

The dimensionless ratio g/ω_c sets the regime: JCM validity requires g/ω_c ≪ 1, typically below a few percent.

How It Is Observed: Rabi Splitting and Collapse-Revival

Two signatures cement the JCM's experimental reality. First, vacuum Rabi splitting: probe the coupled system spectroscopically and a single cavity resonance splits into a doublet separated by 2g. Serge Haroche's group at the École Normale Supérieure demonstrated this with Rydberg atoms in a microwave cavity in the early 1990s; optical and circuit-QED versions followed.

The second, subtler signature is collapse and revival. If the field starts not in a Fock state but in a coherent state |α⟩ (a near-classical field with mean photon number n̄ = |α|²), the atom sees a superposition of many Rabi frequencies Ω_n = 2g√(n+1). These dephase, so the Rabi oscillations collapse — then, because the frequencies form a discrete comb, they re-phase and the oscillations revive at times t_rev ≈ 2π√n̄ / g.

• The collapse time scales as t_c ≈ √2/g (roughly independent of n̄).
• Revival is direct evidence of field quantization — a truly classical field would collapse and never come back.

Rempe, Walther, and Klein observed collapse and revival in a one-atom micromaser in 1987 (Phys. Rev. Lett. 58, 353), a landmark confirmation.

Significance, Cousins, and Open Questions

The Jaynes-Cummings model is the theoretical bedrock of cavity and circuit quantum electrodynamics, and it earned its physics its due: the 2012 Nobel Prize went to Serge Haroche and David Wineland for measuring and manipulating individual quantum systems, work built directly on JCM physics. Today the model underpins superconducting-qubit readout (the dispersive regime, |Δ| ≫ g, shifts the cavity frequency by g²/Δ per photon), single-photon sources, and photon-number-resolving detection.

Its close relatives sharpen its meaning:

• Quantum Rabi model — drop the RWA; needed in the ultrastrong (g/ω_c ≳ 0.1) and deep-strong (g/ω_c ≳ 1) regimes now reached in circuit QED, where the ground state itself contains virtual photons. Daniel Braak gave an analytic solution in 2011.
• Tavis-Cummings model — N atoms sharing one mode, with collective coupling ∝ √N.
• Dicke model — many atoms plus counter-rotating terms, hosting a superradiant phase transition.

Open questions center on driven-dissipative JCM physics, the JCM's own quantum phase transition in the many-photon limit, and pushing coherent single-photon control deeper into the ultrastrong regime.

Jaynes-Cummings vs. the full quantum Rabi model, and coupling regimes across platforms

Aspect	Jaynes-Cummings model	Quantum Rabi model
Interaction term	g(a†σ₋ + aσ₊) — RWA kept	g(a† + a)(σ₊ + σ₋) — full dipole
Conserved quantity	Excitation number N = a†a + σ₊σ₋	Only parity (Z₂ symmetry)
Solvability	Exact, closed-form eigenstates	No elementary closed form (Braak 2011)
Valid when	g ≪ ω_c, ω_a and near resonance	Any coupling, incl. ultrastrong/deep-strong
Counter-rotating terms	Dropped	Retained
Ground state	|g,0⟩, energy −½ħω_a	Squeezed, photon-populated vacuum