---
title: Chiral Dicke Model Overview
url: https://www.emergentmind.com/topics/chiral-dicke-model
type: topic
---

# Chiral Dicke Model Overview

The chiral Dicke model denotes a family of Dicke-type light–matter Hamiltonians in which chirality is encoded either through synthetic gauge flux in coupled cavities or through angular-momentum–selective coupling to multiple cavity modes. In current arXiv usage, the term therefore does not identify a single canonical Hamiltonian but a class of constructions sharing collective light–matter coupling, superradiant ordering, and an additional directional or handed structure. Two representative realizations are the Dicke triangle, where artificial magnetic flux produces a chiral superradiant phase with circulating photon current and tricritical manifolds [2209.00294], and the two-mode chiral Dicke model, where an atomic ensemble couples chirally to two degenerate cavity modes and the Hamiltonian has an exact continuous \(U(1)\) symmetry for arbitrary couplings [2604.21820]. Earlier analyses of the full Dicke model with independent rotating and counter-rotating couplings provide the natural symmetry benchmark for these developments [1011.3075].

## 1. Hamiltonian architectures

One line of development starts from a tricritical Dicke cavity. In the Dicke triangle, each cavity contains an ensemble of \(N\) identical three-level atoms in a \(\Lambda\)-like configuration with levels \(\{|1\rangle,|0\rangle,|-1\rangle\}\), energies \(\{+1,0,-1\}\) in units of \(\Omega\), and cavity photons of frequency \(\omega\). The single-cavity Hamiltonian is
\[
H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},
\]
where the parameter \(\gamma\) controls the relative strength of the two dipole-allowed transitions \(|1\rangle\leftrightarrow|0\rangle\) and \(|0\rangle\leftrightarrow|-1\rangle\), and the effective dimensionless coupling is \(\lambda=g/\sqrt{\Omega\omega}\). Three such cavities are then coupled on an equilateral triangle by photon hopping with a Peierls phase,
\[
H=\sum_{n=1}^3 H_{D,n}+\sum_{n=1}^3 J\left(e^{i\theta}a_n^\dagger a_{n+1}+e^{-i\theta}a_{n+1}^\dagger a_n\right),
\]
with cyclic boundary condition \(a_4\equiv a_1\). The total artificial magnetic flux through the triangular plaquette is, up to gauge, \(3\theta\), so \(\theta\) parameterizes a synthetic magnetic field [2209.00294].

A second construction is the two-mode chiral Dicke model. Its Hamiltonian is
\[
\begin{split}
H_{\rm CDM}=&\;\omega_c(a_1^\dagger a_1+a_2^\dagger a_2)+\omega_z S^z+U S^z(a_1^\dagger a_1+a_2^\dagger a_2)\\
&+\frac{g_1}{\sqrt N}(a_1 S^+ + a_1^\dagger S^-)+\frac{g_2}{\sqrt N}(a_2 S^- + a_2^\dagger S^+),
\end{split}
\]
where mode \(a_1\) couples through co-rotating terms and mode \(a_2\) through counter-rotating terms. The two cavity modes are interpreted as circularly polarized or counter-propagating running-wave modes carrying opposite angular momentum along the cavity axis, and chirality is tied to the handedness of angular-momentum exchange between light and matter [2604.21820].

The single-mode full Dicke model furnishes the antecedent reference point. With a single bosonic mode \(b\), \(N\) two-level atoms, and independent rotating and counter-rotating couplings \(g_1\) and \(g_2\), its Hamiltonian is
\[
H=\omega_0 b^\dagger b+\frac{\Omega}{2}\sum_{j=1}^N \sigma_j^z
+\frac{g_1}{\sqrt N}\sum_{j=1}^N (b\sigma_j^+ + b^\dagger \sigma_j^-)
+\frac{g_2}{\sqrt N}\sum_{j=1}^N (b\sigma_j^- + b^\dagger \sigma_j^+).
\]
This model already distinguishes the rotating-wave, pure counter-rotating, and generic full regimes, and thereby clarifies which symmetry survives when both processes coexist [1011.3075].

## 2. Symmetry content and the meaning of chirality

In the Dicke triangle, chirality is flux-induced. For \(\theta=0,\pi\), the hopping is real and time-reversal symmetry is preserved. For general \(\theta\neq m\pi\), the Hamiltonian is complex, time-reversal symmetry is broken, and the ground state can carry nontrivial circulating photon currents. The model retains a global parity symmetry,
\[
P=\prod_{i=1}^{3}\exp\Big\{i\pi\Big[a_i^\dagger a_i+\sum_{k=1}^N(h^{(k)}+1)\Big]\Big\},
\]
and at \(\theta=0\) the full symmetry includes \(\mathbb{Z}_2\) parity and discrete rotation \(C_3\) [2209.00294].

In the two-mode chiral Dicke model, chirality is angular-momentum selective rather than flux-driven. The defining feature is an exact continuous \(U(1)\) symmetry for all \(g_1\), \(g_2\), and \(U\):
\[
a_1\to e^{i\theta}a_1,\qquad
a_2\to e^{-i\theta}a_2,\qquad
S^\pm\to e^{\mp i\theta}S^\pm .
\]
The symmetry is generated by the conserved quantity
\[
L^z=a_1^\dagger a_1-a_2^\dagger a_2+S^z,
\]
interpreted as the total angular momentum along the cavity axis. The model is therefore chiral in the sense that the two modes carry opposite chirality and angular momentum conservation enforces a robust continuous symmetry rather than a fine-tuned one [2604.21820].

The symmetry distinction from the single-mode full Dicke model is sharp. In the path-integral analysis of the full Dicke model, the rotating-wave limit \(g_2=0\) conserves total excitation number \(N\), the pure counter-rotating limit \(g_1=0\) conserves the excitation difference \(N_-\), and both limits have a continuous \(U(1)\) symmetry. By contrast, when \(g_1\neq 0\) and \(g_2\neq 0\), only parity \(\Pi=e^{i\pi N}\) remains [1011.3075]. This shows that chirality in the two-mode chiral Dicke model is not simply the asymmetry \(g_1\neq g_2\); the robust \(U(1)\) structure requires the two-mode angular-momentum construction itself [2604.21820].

A recurrent misconception is that all chiral Dicke models must break time-reversal symmetry in the same way. The current literature suggests two distinct mechanisms: synthetic-flux chirality, which produces circulating currents and explicitly complex hopping amplitudes, and angular-momentum chirality, which produces exact \(U(1)\) symmetry and a Goldstone sector without requiring a triangular flux geometry.

## 3. Ordered phases and order parameters

For the Dicke triangle, the normal phase is defined by \(\alpha_n=0\) in all cavities. In the superradiant regime one writes
\[
a_n\rightarrow \tilde a_n+\sqrt N\,\alpha_n,\qquad \alpha_n=A_n+iB_n,
\]
so the six real variables \(A_n\) and \(B_n\) serve as order parameters. The conventional superradiant phase has
\[
\alpha_n=A\in\mathbb{R},\qquad \forall n,
\]
so the coherent fields are uniform and real. The chiral superradiant phase instead has complex, nonuniform \(\alpha_n\), and can be represented in a convenient gauge by
\[
A_3=A,\quad A_1=A_2=\tilde A,\quad B_3=0,\quad B_2=-B_1.
\]
More generally,
\[
A_n\neq A_{n+1}=A_{n-1},\qquad B_n=0,\quad B_{n+1}=B_{n-1}.
\]
These configurations break both parity \(\mathbb{Z}_2\) and the threefold rotation \(C_3\), producing a six-fold degenerate ground state after accounting for the three choices of the distinguished cavity and the two parity-related signs [2209.00294].

The signature of chirality in the triangle is the photon current operator
\[
I_{\rm ph}=i\Big[(a_1^\dagger a_2+a_2^\dagger a_3+a_3^\dagger a_1)-\text{h.c.}\Big].
\]
At mean field,
\[
\langle I_{\rm ph}\rangle \propto i\Big[\alpha_1^*\alpha_2+\alpha_2^*\alpha_3+\alpha_3^*\alpha_1-\text{c.c.}\Big].
\]
In the conventional superradiant phase all phase differences vanish and \(\langle I_{\rm ph}\rangle=0\). In the chiral superradiant phase, \(\langle I_{\rm ph}\rangle\neq 0\), its sign changes under \(\theta\to-\theta\), and tuning \(\theta\) drives a jump from zero to finite positive or negative current when entering the CSR region [2209.00294].

The two-mode chiral Dicke model also has a normal and a superradiant phase, but the structure is different. Using a Holstein–Primakoff mapping and coherent-state shifts \(a_i\to \alpha_i+\tilde a_i\), the normal phase is
\[
\alpha_1=\alpha_2=\alpha_3=0.
\]
The superradiant solution has
\[
\alpha_3=|\alpha_3|e^{i\theta},
\]
with an arbitrary phase \(\theta\). The phase boundary is
\[
\sqrt{g_1^2+g_2^2}=g_c=\sqrt{\omega_z\tilde\omega_c},\qquad \tilde\omega_c=\omega_c-\frac{UN}{2},
\]
and for \(U=0\) this reduces to \(g_c=\sqrt{\omega_z\omega_c}\), or equivalently \(g_1^2+g_2^2>\omega_z\omega_c\) in the ordered phase. The order parameter therefore has a magnitude that separates normal and superradiant regimes and a phase that parameterizes the spontaneously broken \(U(1)\) manifold \(S^1\) [2604.21820].

## 4. Critical manifolds, tricriticality, and multiversality

The Dicke triangle supports a richer phase diagram than the ordinary Dicke model. In a single cavity, the tricritical Dicke model already exhibits normal and superradiant phases separated by both first- and second-order lines meeting at a tricritical point. In the triangle, both the conventional superradiant sector and the chiral superradiant sector inherit this structure. The quartic Landau coefficient in both SR and CSR expansions is
\[
c_2=\gamma^2\left(\gamma^2-\frac{1}{2}\right),
\]
so the normal-to-superradiant transition is second order for \(\gamma>1/\sqrt{2}\) and first order for \(\gamma<1/\sqrt{2}\). This yields a line of tricritical points for the SR sector and a line of chiral tricritical points for the CSR sector, both at \(\gamma=1/\sqrt{2}\). The full three-dimensional phase diagram in \((\lambda,\gamma,\theta)\) contains two superradiant regions, SR and CSR, separated by a first-order surface, and at \(\theta=\theta_c\) the normal, SR, and CSR phases meet at triple points [2209.00294].

The critical exponents separate ordinary criticality from tricriticality and also distinguish the uniform and chiral condensation channels. Defining the distance \(L\) from a point to the nearest critical point along a line perpendicular to the relevant critical boundary, the total photon number satisfies
\[
N_{\rm ph}\propto L^\beta.
\]
Along ordinary second-order NP–SR or NP–CSR lines,
\[
\beta_{\rm 2nd}=1,
\]
whereas at both TCP and CTCP,
\[
\beta_{\rm TCP}=\beta_{\rm CTCP}=\frac{1}{2}.
\]
For the lowest excitation gap,
\[
\varepsilon_1\propto L^\eta,
\]
with
\[
\eta_{\rm SR}=\frac{1}{2},\qquad \eta_{\rm CSR}=1.
\]
At the triple points, two excitation energies vanish simultaneously with exponents \(1/2\) and \(1\), reflecting the coexistence of distinct critical modes [2209.00294].

The two-mode chiral Dicke model exhibits a different phenomenon, termed multiversality. The normal and \(U(1)\)-broken superradiant phases are always separated by the same critical circle \(g=g_c\) in the \((g_1,g_2)\) plane, but the gap-closing exponent depends on the approach direction. Writing
\[
g_1=g\cos\phi,\qquad g_2=g\sin\phi,
\]
the lowest normal-phase gap scales generically as
\[
\varepsilon_\Delta\propto |g-g_c|,\qquad z\nu=1.
\]
Along the special line
\[
\cos(2\phi)=-\frac{\omega_c}{\omega_z},
\]
where two excitation branches become degenerate at criticality, the scaling changes to
\[
\varepsilon_\Delta\propto |g-g_c|^{1/2},\qquad z\nu=\frac{1}{2}.
\]
The same pair of phases and the same phase boundary therefore support two distinct universality classes, depending only on the path taken in coupling space [2604.21820].

## 5. Excitations, soft modes, and Goldstone structure

In the Dicke triangle, the normal phase can be reduced, after a Schrieffer–Wolff transformation and projection onto the lowest atomic state, to an effective three-mode bosonic Hamiltonian. Fourier transforming to momenta \(q=0,\pm 2\pi/3\) gives mode energies
\[
\omega_q=\omega-\frac{4g^2\gamma^2}{\Omega}+2J\cos(\theta-q),
\]
and a Bogoliubov transformation yields quasiparticle energies
\[
\varepsilon_q=\frac{1}{2}\left[\omega_q-\omega_{-q}+\sqrt{(\omega_q+\omega_{-q})^2-\frac{64 g^4\gamma^4}{\Omega^2}}\right].
\]
The normal phase is stable as long as all \(\varepsilon_q>0\). Instability of the \(q=0\) mode signals a continuous transition to the uniform SR phase, while instability of the \(q=\pm 2\pi/3\) modes signals condensation into the CSR phase [2209.00294].

In the two-mode chiral Dicke model, quadratic expansion around the mean-field solution yields a Bogoliubov Hamiltonian for the two photon modes and the Holstein–Primakoff boson. In the superradiant phase there are two gapped polaritonic modes with explicitly tunable gaps and, in addition, a Goldstone mode
\[
\varepsilon_\Delta=0
\]
throughout the superradiant phase in the thermodynamic limit. The spectrum is tunable by the chiral imbalance between \(g_1\) and \(g_2\), the detuning \(\omega_z/\omega_c\), and the dispersive coupling \(U\). The paper emphasizes that the structure of one Goldstone mode plus two gapped modes persists for \(U\neq 0\) [2604.21820].

The contrast with the single-mode full Dicke model again clarifies the symmetry content. In the path-integral treatment of the full Dicke model, the rotating-wave and pure counter-rotating limits have a continuous \(U(1)\) symmetry and the superradiant phase contains a zero-energy Goldstone mode. In the generic full model with both \(g_1\neq 0\) and \(g_2\neq 0\), only discrete \(Z_2\) symmetry survives and the superradiant phase has no zero mode; all excitations remain gapped [1011.3075]. This benchmark makes clear why the robust Goldstone mode of the two-mode chiral Dicke model is structurally nontrivial.

The two-mode chiral Dicke model also admits a dissipative mean-field extension. In the supplemental analysis summarized in the paper, a nontrivial steady-state solution exists for \(g_1=g_2=g\) and
\[
g>g_c=\sqrt{\frac{\omega_0}{2\omega}(\kappa^2+\omega^2)},
\]
and the corresponding superradiant \(U(1)\)-broken state is linearly stable. The equilibrium Goldstone interpretation therefore remains relevant, at mean-field level, even in the presence of moderate cavity loss [2604.21820].

## 6. Experimental realizations, related models, and broader scope

The Dicke triangle was proposed as implementable both with atoms in optical cavities and in circuit QED. In the optical-cavity setting, three cavities or three sites of a cavity array form a triangle, each cavity hosts a large ensemble of atoms with a suitable three-level structure, photon hopping is engineered through evanescent coupling or fiber links, and time-periodic modulation of hopping generates the synthetic phase \(\theta\). In circuit QED, three superconducting resonators are coupled in a loop, each resonator is coupled to many effective three-level systems, and parametrically modulated couplers implement synthetic gauge fields. Superradiance can be detected through cavity transmission spectra, homodyne detection, or quadrature measurements; chirality can be accessed through direction-dependent output or phase correlations, with \(I_{\rm ph}\) expected to jump discontinuously across the SR–CSR boundary and hysteresis expected for first-order transitions [2209.00294].

The two-mode chiral Dicke model is motivated by platforms with two counter-propagating or two circularly polarized modes. The paper identifies high-finesse ring cavities supporting two running-wave modes, circuit-QED realizations with two resonator modes and a superconducting qubit ensemble, and chiral quantum-optics architectures such as waveguide QED or photonic-crystal waveguides with spin–orbit coupling of light as candidate settings. Observation of the ordered phase would require detecting macroscopic occupations \(|\langle a_1\rangle|\), \(|\langle a_2\rangle|\sim \sqrt N\), a freely rotating relative phase constrained by the \(U(1)\) symmetry, and a low-energy near-zero-frequency spectral line associated with the Goldstone mode. Observation of multiversality would require controllable variation of the ratio \(g_1/g_2\) and spectroscopic resolution of \(|g-g_c|\) versus \(|g-g_c|^{1/2}\) gap closing [2604.21820].

The broader literature contains adjacent chiral Dicke-like constructions. In chiral waveguide optomechanics, atoms trapped near a chiral ring waveguide map to a generalized quantum Rabi model, and the three-atom case realizes a \(\mathbb{Z}_3\)-symmetric Dicke-like problem with a first-order quantum phase transition and multicomponent Schrödinger-cat ground states [2009.01289]. In quantum Hall systems, the coupling of nuclear spins to a linearly dispersing Nambu–Goldstone mode in the canted antiferromagnetic phase yields an effective Dicke model with a continuous-mode bosonic field, where the long-wavelength approximation produces collective coupling of the total nuclear spin to the NG mode [1510.04792]. These constructions are not identical to the flux-threaded Dicke triangle or the two-mode chiral Dicke model, but they show that chiral Dicke physics extends beyond conventional cavity-QED geometries.

Several open directions are explicitly identified. For the Dicke triangle these include larger rings and \(2\)D arrays under synthetic flux, quantum fluctuations beyond the \(N\to\infty\) mean-field limit, real-time dynamics across TCP and CTCP, and the inclusion of drive and dissipation [2209.00294]. For the two-mode chiral Dicke model they include a full driven-dissipative phase diagram, periodic and quasiperiodic driving, finite temperature, disorder, multimode or multiband generalizations, and the effect of chirality and the robust Goldstone mode on quantum chaos and scrambling [2604.21820]. Taken together, these proposals suggest that the chiral Dicke model is best viewed as a developing family of long-range light–matter theories in which symmetry, chirality, and superradiance are intertwined in ways that are not available in the standard single-mode Dicke model.

Source: https://www.emergentmind.com/topics/chiral-dicke-model