Dicke Model

One-Sentence Description

The Dicke model couples many identical two-level systems collectively to a single quantized bosonic mode.

Physical Setup

Consider $N$ two-level emitters interacting with the same cavity or field mode. When the emitters are equivalent and couple symmetrically, collective spin operators compress the many-emitter Hilbert space and reveal enhanced coupling effects.

The model is used for superradiance, collective strong coupling, cavity QED, and quantum phase-transition idealizations. Experimental modeling often also needs drive, loss, inhomogeneity, and gauge constraints.

Hilbert Space

The full Hilbert space is

$$\mathcal H = \left(\mathbb C^2\right)^{\otimes N} \otimes \mathcal F_B.$$

Define collective spin operators

$$J_\alpha = \frac12 \sum_{j=1}^{N}\sigma_j^\alpha.$$

Symmetric initial states and symmetric couplings often restrict the dynamics to a smaller collective-spin sector.

Hamiltonian

A common closed Dicke-model convention is

$$H = \hbar\omega_c a^\dagger a + \hbar\omega_0 J_z + \frac{2\hbar g}{\sqrt N} \left(a+a^\dagger\right)J_x.$$

The factor $1/\sqrt N$ is a thermodynamic scaling convention. A rotating-wave relative, often called the Tavis-Cummings model, uses

$$H_{\mathrm{TC}} = \hbar\omega_c a^\dagger a + \hbar\omega_0 J_z + \frac{\hbar g}{\sqrt N} \left( aJ_+ + a^\dagger J_- \right).$$

Parameters

Symbol	Meaning
$N$	number of two-level systems
$\omega_c$	common bosonic-mode frequency
$\omega_0$	emitter transition frequency
$g$	collective coupling scale
$J_\alpha$	collective spin component

Solvability

The model is highly structured because collective spin symmetry reduces the effective Hilbert space in symmetric sectors. It is not generically a finite collection of independent Jaynes–Cummings models: all emitters couple through the same mode.

In thermodynamic idealizations, the Dicke model has a superradiant phase-transition structure. In microscopic light-matter systems, gauge invariance, diamagnetic terms, losses, and finite-size effects must be handled before interpreting that transition literally.

Key Observables

• Collective spin polarization.
• Photon number.
• Superradiant emission intensity.
• Normal-mode splitting.
• Correlations between emitters mediated by the common mode.

What It Teaches

The Dicke model teaches collective enhancement, permutation symmetry, bright and dark states, and the difference between single-emitter cavity physics and many-emitter collective dynamics.

It also warns that a compact Hamiltonian can hide substantial physical assumptions about gauge choice, mode truncation, and dissipation.

Canonical Links

• Jaynes–Cummings Model
• Rabi Model
• Two-Level System
• Entanglement in Quantum Optics
• Bosonic Commutation Relations
• AMO Model Index

Common Mistakes

• Confusing Dicke superradiant emission with the equilibrium Dicke-model phase transition.
• Forgetting the collective coupling convention and its $N$ scaling.
• Treating every multi-emitter cavity model as fully symmetric.
• Ignoring gauge and diamagnetic-term caveats in ultrastrong coupling discussions.

Quick Check

Why can symmetric Dicke dynamics be much smaller than the full $2^N$ emitter Hilbert space?

Solution

If all emitters couple identically and the initial state is symmetric, the dynamics can remain in a collective-spin sector with dimension $N+1$ rather than the full $2^N$ spin space. This reduction depends on symmetry assumptions.

References

• R. H. Dicke, “Coherence in spontaneous radiation processes,” Physical Review 93, 99-110, 1954.
• M. Tavis and F. W. Cummings, “Exact solution for an N-molecule-radiation-field Hamiltonian,” Physical Review 170, 379-384, 1968.
• K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field,” Annals of Physics 76, 360-404, 1973.