The Jaynes–Cummings model (JCM) is a theoretical framework in quantum optics that describes the interaction between a single two‑level atomic system (or qubit) and a single mode of a quantized electromagnetic field, typically within a high‑Q cavity. Introduced independently by Edwin T. Jaynes and Fred W. Cummings in 1963, the model provides a solvable example of light–matter coupling and forms a cornerstone for understanding phenomena such as vacuum Rabi oscillations, collapse and revival of atomic inversion, and photon blockade.
Hamiltonian
In the rotating‑wave approximation (RWA), the JCM Hamiltonian (ℏ = 1) is expressed as
$$ \hat{H} = \omega_c \hat{a}^{\dagger}\hat{a} + \frac{\omega_a}{2}\hat{\sigma}z + g\left(\hat{a}^{\dagger}\hat{\sigma}- + \hat{a}\hat{\sigma}_+\right), $$
where
- $\omega_c$ is the frequency of the cavity mode,
- $\omega_a$ is the transition frequency of the two‑level atom,
- $\hat{a}^{\dagger}$ and $\hat{a}$ are the photon creation and annihilation operators,
- $\hat{\sigma}z$, $\hat{\sigma}\pm$ are Pauli operators acting on the atomic state, and
- $g$ is the atom‑field coupling strength.
The RWA neglects counter‑rotating terms $\hat{a}\hat{\sigma}-$ and $\hat{a}^{\dagger}\hat{\sigma}+$ that oscillate rapidly compared with the interaction timescale, an approximation valid when $g \ll \omega_c, \omega_a$.
Exact solution
Because the total excitation number $\hat{N} = \hat{a}^{\dagger}\hat{a} + \frac{1}{2}(\hat{\sigma}_z + 1)$ commutes with $\hat{H}$, the Hilbert space decomposes into invariant subspaces labelled by the integer $n$ (photon number). Within each subspace, the Hamiltonian reduces to a $2\times 2$ matrix whose eigenvalues are
$$ E_{n,\pm} = \left(n+\tfrac{1}{2}\right)\omega_c \pm \tfrac{1}{2}\sqrt{\Delta^{2}+4g^{2} (n+1)}, $$
with detuning $\Delta = \omega_a - \omega_c$. The corresponding eigenstates are known as dressed states or polaritons.
Key phenomena
- Vacuum Rabi oscillations – Coherent exchange of excitation between atom and field at the Rabi frequency $\Omega_n = \sqrt{\Delta^{2}+4g^{2}(n+1)}$.
- Collapse and revival – For an initial coherent field state, the atomic inversion exhibits a rapid collapse due to dephasing among different $n$-dependent Rabi frequencies, followed by partial revivals at characteristic times $t_{\text{rev}} \approx 2\pi\sqrt{\bar{n}}/g$, where $\bar{n}$ is the mean photon number.
- Quantum entanglement – The interaction generates entangled atom‑field states, serving as a prototype for quantum information protocols.
Extensions and generalizations
- Multi‑atom (Dicke) models – Incorporating multiple two‑level systems leads to the Tavis‑Cummings model.
- Beyond the RWA – Inclusion of counter‑rotating terms yields the full quantum Rabi model, which is relevant in ultra‑strong coupling regimes where $g$ approaches $\omega_c$ or $\omega_a$.
- Driven and dissipative versions – Adding external driving fields and coupling to reservoirs enables the study of cavity QED with loss, decoherence, and steady‑state phenomena.
Experimental realizations
The JCM has been implemented in a variety of physical platforms, including:
- Microwave cavity QED with Rydberg atoms (Haroche group).
- Optical cavity QED with trapped neutral atoms or ions.
- Circuit QED, where superconducting qubits play the role of artificial atoms coupled to microwave resonators.
- Quantum dot or nitrogen‑vacancy‑center systems embedded in photonic crystal cavities.
These experiments have verified many of the model’s predictions, such as discrete vacuum Rabi splitting observed in transmission spectra and time‑resolved collapse‑revival dynamics.
Significance
The Jaynes–Cummings model provides a minimal yet exactly solvable description of quantized light–matter interaction. Its analytical tractability makes it a pedagogical tool for illustrating quantum coherence, entanglement, and measurement back‑action. Moreover, it underpins the design and analysis of modern quantum technologies, including quantum memories, single‑photon sources, and quantum transducers.