Dicke Model in Quantum Optics

• The Dicke model is a foundational theory in quantum optics describing N two-level systems interacting with a single bosonic mode, highlighting phenomena like quantum chaos and phase transitions.
• Its anisotropic extensions and integrable limits reveal rich dynamics, where tuning coupling parameters yields transitions between Poissonian and Wigner-Dyson level spacing statistics.
• Analytical and numerical techniques, including the Bethe Ansatz and OTOC diagnostics, offer practical insights into ergodicity-breaking transitions and the superradiant quantum phase transition.

The Dicke model is a canonical theory in quantum optics and many-body physics describing the collective interaction of $N$ two-level systems (spins or atoms) with a single bosonic (photonic) mode. It serves as a paradigmatic platform for studying quantum phase transitions, quantum chaos, integrability-breaking phenomena, and the emergence of macroscopic order from microscopic interactions. The model encompasses a range of behaviors—integrable limits, quantum chaotic regimes, excited-state quantum phase transitions, and allows for generalization to multi-level systems, disorder, and open-system dynamics.

1. The Dicke Model and Its Anisotropic Extensions

The standard Dicke Hamiltonian for $N$ two-level systems with energy splitting $\omega_0$ coupled homogeneously to a single bosonic mode of frequency $\omega$ is: $$H = \omega\,a^\dagger a + \omega_0\,J_z + \frac{g}{\sqrt{2j}}\,(a^\dagger J_- + a J_+) + \frac{g'}{\sqrt{2j}}\,(a^\dagger J_+ + a J_-)$$ where $a,a^\dagger$ are bosonic mode operators, $J_z, J_\pm$ are collective spin-$j$ operators ($j=N/2$), $g$ is the coupling for co-rotating (energy-conserving) terms, and $g'$ is the coupling for counter-rotating (energy non-conserving) terms. The standard or "isotropic" Dicke model is recovered for $g=g'$, while $g'=0$ yields the rotating-wave Tavis-Cummings limit. This model preserves a discrete parity symmetry, and its eigenstates are classified by total parity $\Pi = \exp[i\pi(a^\dagger a + J_z + j)]$ (Buijsman et al., 2016, Alexanian, 16 Jul 2025).

The anisotropic Dicke model (ADM) allows tuning $g$ and $g'$ independently: $$H_{\mathrm{ADM}} = \omega\,a^\dagger a + \omega_0\,J_z + \frac{g_1}{\sqrt{2j}}(a^\dagger J_- + a J_+) + \frac{g_2}{\sqrt{2j}}(a^\dagger J_+ + a J_-)$$ with $g_1 = g$, $g_2 = g'$. This generalization reveals a rich tapestry of integrable and non-integrable regimes, quantum phase transition behaviors, and ergodicity-breaking phenomena (Buijsman et al., 2016).

2. Integrable Limits and Proximity to Integrability

Integrability is realized in two limits: (i) $g_2=0$, which yields the rotating-wave Tavis-Cummings/Gaudin model with $U(1)$ excitation number conservation; and (ii) $g_1=0$, a case related by spin rotation, also conferring integrability. In these limits, the system admits a complete set of commuting conserved quantities and solvable structure (Bethe Ansatz), with spectral statistics characterized by Poissonian level spacing distributions (Buijsman et al., 2016). Away from these axes, the model generally becomes nonintegrable.

A key finding is the presence of "quasi-integrable tongues": extended non-ergodic regions in parameter space emanating from either $g_1=0$ or $g_2=0$. The extent of these tongues is controlled by proximity to the integrable axes, forming a quantum counterpart to the Kolmogorov-Arnold-Moser (KAM) phenomenon: small perturbations from integrability do not immediately destroy regularity or integrals of motion (Buijsman et al., 2016).

A separate semiclassical/Born–Oppenheimer approach demonstrates that, even when the full Dicke model is non-integrable, an approximate second integral of motion exists in the superradiant phase and below a chaos threshold. The adiabatic invariant

$$I = J_{z'} = \frac{J_z + \alpha(a+a^\dagger)J_x}{\sqrt{1+\alpha^2(a+a^\dagger)^2}}$$

(with explicit $\alpha$) organizes the regular part of the spectrum into bands labeled by "$m'$" (Relaño et al., 2016).

3. Ergodicity, Quantum Chaos, and Dynamical Diagnostics

The ergodic–non-ergodic transition (ENET) in the Dicke model is sharply demarcated by the aforementioned proximity to integrable limits. Two principal diagnostics are employed:

• Level Spacing Statistics: For ordered eigenvalues $\{E_n\}$, spacings $s_n = E_{n+1} - E_n$ display Poisson statistics ($P_P(s)=e^{-s}$) in integrable regions and Wigner-Dyson (GOE) statistics ($P_{WD}(s)=\frac{\pi}{2}s e^{-\pi s^2/4}$) in quantum chaotic (ergodic) regions. The average ratio of consecutive spacings $$\langle r \rangle = \langle \min(s_n, s_{n-1})/\max(s_n, s_{n-1})\rangle$$ exhibits a jump from $\approx0.386$ (Poisson) to $\approx0.5307$ (GOE) as the system is tuned away from integrable axes (Buijsman et al., 2016).
• Quantum Butterfly/OTOC (Out-of-Time-Ordered Correlator): For operators $V, W$,

$$F(t) = \frac{1}{2} \langle V^\dagger(0) W^\dagger(t) V(0) W(t) + \mathrm{h.c.} \rangle_\beta$$

In the ergodic regime, $1 - F(t)/F(0)$ grows and saturates, reflecting rapid operator spreading (scrambling). In non-ergodic regions, $F(t)$ remains close to its initial value (Buijsman et al., 2016).

Both diagnostics map the (co-rotating, counter-rotating) coupling plane and reveal extended non-ergodic regions that have no special relation to the thermodynamic normal–superradiant quantum phase transition.

4. Quantum Phase Transitions and Their Independence from ENET

The Dicke model exhibits a second-order quantum phase transition (QPT) between normal and superradiant phases, defined (for $j\to\infty$) by

$g_1+g_2 = \sqrt{\omega\,\omega_0}$

with superradiant behavior for $\langle a^\dagger a\rangle/j \sim O(1)$. However, the ergodic–non-ergodic transition does not coincide with this QPT: no singularity or qualitative change in ergodic diagnostics occurs as the QPT is crossed. Instead, the ENET is governed strictly by proximity to the integrable axes, confirming the decoupling of quantum chaos and equilibrium QPT phenomena in this model (Buijsman et al., 2016).

5. Static and Dynamical Phase Diagrams

Numerical studies for $j=10$ ($N=20$ spins) and $\omega=\omega_0=1$ map phase diagrams in the $(g_1,g_2)$ plane:

Diagnostic	Ergodic Regime	Non-Ergodic Regime
$\langle r \rangle$	$\approx 0.53$ (GOE)	$\approx 0.39$ (Poisson)
OTOC map	Large contrast/scrambling	Small contrast, slow decay
QPT boundary	$g_1+g_2=1$ (no ENET sign.)

The boundary between ergodic and integrable-like dynamics approximates contours of constant minimal distance to $g_1=0$ or $g_2=0$. For $g_2=0$ and $g_1$ increasing, ergodic behavior appears only for $g_1 \gtrsim 1.0$, while on the isotropic line, the crossover appears near $g \simeq 0.5$ (Buijsman et al., 2016).

6. Broader Implications and Quantum KAM Scenario

The observation that the Dicke model’s ergodic–non-ergodic boundary precisely tracks its distance to integrable limits is analogous, at the quantum level, to the KAM theorem for classical Hamiltonian systems: integrability-breaking terms must exceed a threshold before chaotic behavior dominates. In the quantum Dicke context, the entire spectrum can remain regular (nonergodic) for parameter ranges sufficiently close to integrability, regardless of QPT lines (Buijsman et al., 2016). This hints toward a universal scenario for quantum systems with classical analogues.

7. Analytical and Numerical Tools

For practical and theoretical investigations, analytical methods (adiabatic/Born–Oppenheimer separation, Bethe Ansatz in integrable limits) are augmented by matrix diagonalization and spectral analysis. The approximate integrability in regular (“pre-chaotic”) regions enables semiclassical quantization, construction of adiabatic invariants, and explicit calculation of expectation values and observables throughout broad parameter regimes (Relaño et al., 2016). Principal numerical benchmarks include analysis of Peres lattices, computation of principal component numbers (NPC), and scaling analysis of errors versus system size.

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References:

• Non-ergodicity in the Anisotropic Dicke model (Buijsman et al., 2016)
• Approximated integrability of the Dicke model (Relaño et al., 2016)