Jaynes–Cummings Model

The Jaynes–Cummings model describes a two-level system coupled to a single quantized oscillator or cavity mode after applying a rotating-wave approximation. It is a standard model for cavity quantum electrodynamics, quantum optics, and light-matter coupling.

Formula Hook

A common Hamiltonian is

$$H_{\mathrm{JC}} = \hbar\omega_c a^\dagger a + \frac{\hbar\omega_0}{2}\sigma_z + \hbar g \left( a\sigma_+ + a^\dagger\sigma_- \right).$$

The interaction exchanges one excitation between the mode and the two-level system. The conserved excitation number is

$$N = a^\dagger a + \sigma_+\sigma_-.$$

The counter-rotating terms $a\sigma_-$ and $a^\dagger\sigma_+$ are omitted in the rotating-wave approximation.

Canonical Home

The canonical derivation, exact spectrum, propagator, vacuum Rabi dynamics, and collapse and revival are developed in the Jaynes–Cummings Model. This page remains a short glossary hook. For ingredients, see Two-Level Systems, Harmonic Oscillator, and Rotating-Wave Approximation.

Common Confusions

• The model is not the full light-matter Hamiltonian; it is an effective near-resonant model.
• The rotating-wave approximation has a regime of validity and can fail in ultrastrong coupling or far off resonance.
• The two-level approximation must be justified for the atom, qubit, or transition being modeled.
• The coupling constant $g$ depends on normalization, mode volume, dipole matrix elements, and convention.

Related Entries

• Hamiltonian
• Superposition
• Harmonic Oscillator

References

• E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proceedings of the IEEE 51, 89-109, 1963.
• B. W. Shore and P. L. Knight, “The Jaynes-Cummings model,” Journal of Modern Optics 40, 1195-1238, 1993.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.