Jaynes–Cummings Hamiltonian

Hamiltonian

A common Jaynes–Cummings Hamiltonian is

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

It conserves the total excitation number

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

Assumptions

• A two-level system couples to one bosonic mode.
• The rotating-wave approximation has been made.
• Coupling is near resonant and weak enough for the model regime.
• Counter-rotating terms are omitted.

Common Mistakes

• Treating the Jaynes–Cummings Hamiltonian as exact far from resonance or in ultrastrong coupling.
• Forgetting the rotating-wave approximation.
• Confusing $\omega_c$ and $\omega_0$.
• Dropping tensor-product identities when building matrices.

Canonical Links

• Canonical Jaynes–Cummings Model
• Jaynes–Cummings Model
• Interactions and Coupling Terms
• Creation and Annihilation Operators
• Two-Level System Hamiltonian

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.