Rabi Model

One-Sentence Description

The quantum Rabi model couples one two-level system to one quantized oscillator mode while retaining both rotating and counter-rotating interaction terms.

Physical Setup

The model describes a minimal closed light-matter system: a two-level atom, artificial atom, spin, or qubit interacting with a single cavity or oscillator mode. It is the natural parent model of Jaynes–Cummings when the rotating-wave approximation is not made.

Hilbert Space

The Hilbert space is a tensor product

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

where $\mathcal F_B$ is the Fock space of one bosonic oscillator mode.

Hamiltonian

A common convention is

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

Equivalently, since $\sigma_x=\sigma_++\sigma_-$, the interaction contains

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

The last two terms are counter-rotating terms. They are the terms removed in the Jaynes–Cummings rotating-wave approximation.

Parameters

Symbol	Meaning
$\omega_c$	oscillator or cavity-mode frequency
$\omega_0$	two-level transition frequency
$g$	light-matter coupling strength
$a,a^\dagger$	bosonic annihilation and creation operators
$\sigma_\alpha$	Pauli operators on the two-level system

Solvability

The quantum Rabi model has a conserved parity symmetry, but it does not conserve excitation number. It is exactly solvable in a specialized analytic sense and is also widely treated numerically by truncating the oscillator basis with convergence checks.

When $g$ is weak and the system is near resonance, the rotating-wave approximation reduces the model to Jaynes–Cummings.

Key Observables

• Dressed eigenenergies.
• Photon-number distribution.
• Two-level inversion.
• Parity sectors.
• Bloch-Siegert shift from counter-rotating terms.

What It Teaches

The Rabi model teaches the difference between semiclassical Rabi oscillations, quantum oscillator coupling, and rotating-wave effective models. It is especially important in strong and ultrastrong coupling regimes where counter-rotating terms are not negligible.

Canonical Links

• Jaynes–Cummings Model
• Rotating-Wave Approximation
• Two-Level System
• Creation and Annihilation Operators
• Interactions and Coupling Terms
• AMO Model Index

Common Mistakes

• Using Jaynes–Cummings formulas after leaving the rotating-wave regime.
• Assuming excitation number is conserved in the full Rabi model.
• Confusing classical driven Rabi oscillations with the quantized Rabi model.
• Interpreting oscillator-basis truncation results without convergence checks.

Quick Check

Which terms in the Rabi interaction fail to conserve the Jaynes–Cummings excitation number?

Solution

The counter-rotating terms $a\sigma_-$ and $a^\dagger\sigma_+$ change the oscillator and two-level excitation in the same direction, so they change the total excitation number by two units.

References

• I. I. Rabi, “On the process of space quantization,” Physical Review 49, 324-328, 1936.
• I. I. Rabi, “Space quantization in a gyrating magnetic field,” Physical Review 51, 652-654, 1937.
• D. Braak, “Integrability of the Rabi model,” Physical Review Letters 107, 100401, 2011.