Jaynes–Cummings Model

One-Sentence Description

The Jaynes–Cummings model is the rotating-wave model of one two-level system coherently exchanging excitations with one quantized oscillator or cavity mode.

Physical Setup

The model applies when a two-level transition is near resonance with a single quantized mode and the coupling is weak enough that rapidly rotating counter-rotating terms can be neglected. It is a standard ideal model for cavity QED, circuit QED, trapped-ion sideband analogies, and quantum optics. The physical mode normalization, loss-rate dictionary, cooperativity, and Purcell regimes are developed in Cavity QED. The exact block-diagonal solution, propagator, and collapse-and-revival dynamics are developed at the canonical Jaynes–Cummings Model.

Hilbert Space

The Hilbert space is

$$\mathcal H = \mathbb C^2 \otimes \mathcal F_B.$$

A convenient basis is $\lvert g,n\rangle$ and $\lvert e,n\rangle$, where $g,e$ label the two-level system and $n$ labels oscillator occupation.

Hamiltonian

A common convention is

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

It conserves

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

This conservation splits the problem into two-dimensional blocks for each nonzero excitation number.

Parameters

Symbol	Meaning
$\omega_c$	oscillator or cavity frequency
$\omega_0$	two-level transition frequency
$\Delta=\omega_0-\omega_c$	detuning
$g$	coupling strength
$n$	oscillator occupation label

Solvability

The model is exactly solvable because total excitation number is conserved. In the subspace spanned by $\lvert e,n\rangle$ and $\lvert g,n+1\rangle$, the dressed-state splitting is controlled by

$$\Omega_n = \sqrt{\Delta^2+4g^2(n+1)}.$$

At resonance, the splitting scales as $\sqrt{n+1}$.

Key Observables

• Vacuum Rabi splitting.
• Dressed eigenstates.
• Excited-state population inversion.
• Photon-number-dependent oscillation frequency.
• Collapse and revival in non-number-state fields.

What It Teaches

The Jaynes–Cummings model teaches quantized excitation exchange, dressed states, vacuum Rabi oscillations, and the operational meaning of a rotating-wave approximation.

It is also a compact example of how a tensor-product interaction entangles a two-level system with a bosonic mode.

Canonical Links

• Canonical Jaynes–Cummings Model
• Jaynes–Cummings Hamiltonian
• Jaynes–Cummings Glossary Entry
• Rabi Model
• Dressed States
• Rotating-Wave Approximation
• Interactions and Coupling Terms

Common Mistakes

• Forgetting that the model already assumes the rotating-wave approximation.
• Using the conserved excitation number for the full Rabi model.
• Confusing detuning $\Delta$ with the dressed splitting $\Omega_n$.
• Omitting cavity loss or spontaneous emission when comparing to an open experiment.

Quick Check

Why does the Jaynes–Cummings Hamiltonian couple $\lvert e,n\rangle$ to $\lvert g,n+1\rangle$?

Solution

The term $a^\dagger\sigma_-$ lowers the two-level system from excited to ground while creating one oscillator quantum. The total excitation number is unchanged.

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.