The fundamental building block of light-matter interaction is the coherent exchange of a single quantum of energy between a single discrete two-level atom and a single quantized electromagnetic cavity mode. Formulated in 1963 by Edwin Jaynes and Fred Cummings, the Jaynes-Cummings Model is the quintessential solvable paradigm of Cavity Quantum Electrodynamics (Cavity QED). In the strong-coupling regime, atomic states and photon states hybridize into entangled dressed states, exhibiting reversible vacuum Rabi oscillations.
1. The Dipole Interaction and Rotating Wave Approximation (RWA)
Consider a two-level atom with ground state (|g
angle) and excited state (|e
angle) separated by transition frequency (omega_a) ((hat{H}_{ ext{atom}} = rac{1}{2}hbar omega_a hat{sigma}_z)), coupled via electric dipole interaction (hat{V} = -hat{mathbf{d}} cdot hat{mathbf{E}}) to a single cavity mode of frequency (omega_c) ((hat{H}_{ ext{field}} = hbar omega_c hat{a}^dagger hat{a})).
Expressing the dipole operator via atomic Pauli ladder operators (hat{mathbf{d}} = mathbf{d}_{eg} (hat{sigma}_+ + hat{sigma}_-)) and the field via annihilation/creation operators, the interaction Hamiltonian contains four distinct operator products:
[
hat{V} = hbar g (hat{sigma}_+ + hat{sigma}_-) (hat{a} + hat{a}^dagger) = hbar g left( hat{a} hat{sigma}_+ + hat{a}^dagger hat{sigma}_- + hat{a} hat{sigma}_- + hat{a}^dagger hat{sigma}_+
ight)
]
where (g = -rac{mathbf{d}_{eg} cdot boldsymbol{psilon}}{hbar}sqrt{rac{hbar omega_c}{2 psilon_0 V}}) is the single-photon vacuum Rabi coupling frequency.
Near resonance ((Delta = omega_a – omega_c ll omega_a + omega_c)), terms in the interaction picture rotate at differing frequencies: the energy-conserving terms (hat{a} hat{sigma}_+) (absorbing a photon while exciting the atom) and (hat{a}^dagger hat{sigma}_-) (emitting a photon while de-exciting the atom) oscillate slowly at (pm Delta), whereas the counter-rotating terms (hat{a} hat{sigma}_-) and (hat{a}^dagger hat{sigma}_+) oscillate at rapid frequencies (pm (omega_a + omega_c)). Neglecting counter-rotating terms constitutes the Rotating Wave Approximation (RWA), yielding the celebrated Jaynes-Cummings Hamiltonian:
[
hat{H}_{ ext{JC}} = hbar omega_c hat{a}^dagger hat{a} + rac{1}{2}hbar omega_a hat{sigma}_z + hbar g (hat{a} hat{sigma}_+ + hat{a}^dagger hat{sigma}_-)
]
2. Conservation of Excitation and Block Diagonalization
The total excitation number operator (hat{N} = hat{a}^dagger hat{a} + rac{1}{2}(hat{sigma}_z + hat{I})) commutes identically with the Jaynes-Cummings Hamiltonian: ([hat{H}_{ ext{JC}}, hat{N}] = 0). Consequently, (hat{H}_{ ext{JC}}) block-diagonalizes into independent (2 imes 2) subspaces spanned by the uncoupled product states ({|e, n
angle, |g, n+1
angle}) for each manifold (n ge 0).
Within the (n)-th subspace, the Hamiltonian matrix takes the form:
[
hat{H}_n = hbar begin{pmatrix} (n + rac{1}{2})omega_c + rac{Delta}{2} & gsqrt{n+1} gsqrt{n+1} & (n + rac{1}{2})omega_c – rac{Delta}{2} nd{pmatrix}
]
3. Dressed States and Vacuum Rabi Splitting
Diagonalizing (hat{H}_n) yields the hybridized atom-field dressed states (|+, n
angle) and (|-, n
angle):
[
|+, n
angle = cos heta_n |e, n
angle + sin heta_n |g, n+1
angle
]
[
|-, n
angle = -sin heta_n |e, n
angle + cos heta_n |g, n+1
angle
]
where the mixing angle is ( an(2 heta_n) = 2gsqrt{n+1}/Delta). The eigenenergies are:
[
E_{pm, n} = left(n + rac{1}{2}
ight)hbar omega_c pm rac{hbar}{2}Omega_n(Delta)
]
where (Omega_n(Delta) = sqrt{Delta^2 + 4 g^2 (n+1)}) is the generalized Rabi frequency.
At exact resonance ((Delta = 0)) for (n = 0), the unperturbed degenerate states (|e, 0
angle) and (|g, 1
angle) split into symmetric and antisymmetric superpositions separated by the energy gap (Delta E = 2hbar g). This spectroscopic doublet is the Vacuum Rabi Splitting, proving that the vacuum state itself exerts a physical dynamical coupling.
4. Vacuum Rabi Oscillations and Strong Coupling Regimes
If an excited atom is placed into a pristine vacuum cavity at (t = 0) ((|psi(0)
angle = |e, 0
angle)), its temporal state evolution oscillates deterministically:
[
|psi(t)
angle = cos(gt) |e, 0
angle – i sin(gt) |g, 1
angle
]
The probability of finding the atom in its excited state oscillates sinusoidally:
[
P_e(t) = |langle e, 0 | psi(t)
angle|^2 = cos^2(gt) = rac{1 + cos(2gt)}{2}
]
Observing these coherent oscillations requires entering the strong-coupling regime, where the coherent interaction rate (g) dominates both atomic spontaneous decay into unguided modes ((gamma)) and photon cavity leakage rate ((kappa)):
[
g gg (kappa, gamma)
]
Achieved in superconducting circuit QED (transmon qubits in microwave stripline resonators) and Rydberg atoms traversing superconducting Fabry-Pérot cavities (Serge Haroche, 2012 Nobel Prize), the Jaynes-Cummings dynamics provides the foundational gate architecture for quantum information processing.
