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arXiv:2607.21583

Complexity transition in the Dicke model of light-matter interaction

Yicheng Zhang, Erhai Zhao

cond-mat.quant-gasquant-phSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations
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Abstract

Tuning the coupling strength $g$ of an interacting quantum system may drive a sudden change in its ground-state or thermal properties. To identify and grasp non-analytical, or even discontinuous, transitions in far-from-equilibrium dynamics proves more challenging. Recently Krylov complexity $C_K$ has offered fresh insights about operator growth, thermalization, and chaos in quantum dynamics. Yet it remains unclear if, and how, changing $g$ can trigger a sharp transition in the complexity measures. Here we present evidence for such a transition by mapping out the complexity phase diagram of the paradigmatic Dicke model describing two-level atoms coupled to a cavity photon mode. Two qualitatively different regimes of dynamics are identified and characterized. At the transition, the slope of $C_K$ changes suddenly to coincide with a jump in the Krylov entropy. We elucidate the nature of the regime change from the wave packet dynamics in Krylov space, where a particle is confined by a roughly linear potential but hops as if it lives in a Rindler reference frame. The competition between confinement, which leads to bouncing, and deconfinement by Rindler hopping, which leads to the destruction of wave packet analogous to gravitational spaghettification, is sensitive to the disorder in Lanczos coefficients. The framework outlined here can be applied to other quantum many-body systems.

Comments
12 pages, 12 figures

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Tuning the coupling strength ggg of an interacting quantum system may drive a sudden change in its ground-state or thermal properties. To identify and grasp non-analytical, or even discontinuous, transitions in far-from-equilibrium dynamics proves more challenging. Recently Krylov complexity CKC_KCK​ has offered fresh insights about operator growth, thermalization, and chaos in quantum dynamics. Yet it remains unclear if, and how, changing ggg can trigger a sharp transition in the complexity measures. Here we present evidence for such a transition by mapping out the complexity phase diagram of the paradigmatic Dicke model describing two-level atoms coupled to a cavity photon mode. Two qualitatively different regimes of dynamics are identified and characterized. At the transition, the slope of CKC_KCK​ changes suddenly to coincide with a jump in the Krylov entropy. We elucidate the nature of the regime change from the wave packet dynamics in Krylov space, where a particle is confined by a roughly linear potential but hops as if it lives in a Rindler reference frame. The competition between confinement, which leads to bouncing, and deconfinement by Rindler hopping, which leads to the destruction of wave packet analogous to gravitational spaghettification, is sensitive to the disorder in Lanczos coefficients. The framework outlined here can be applied to other quantum many-body systems.
Control and manipulation of quantum engineered systems allows for the utilization of time-dependent parametric modulations for accessing novel out-of-equilibrium phenomena. In the absence of such driving, the dissipative Dicke model exhibits a fascinating out-of-equilibrium many-body phase transition as a function of a coupling between a driven photonic cavity and numerous two-level atoms. We study the effect of a parametric modulation of this coupling and discover a rich phase diagram as a function of the modulation strength. We find that in addition to the established normal and super-radiant phases, a new phase with pulsed superradiance, which we term dynamical normal phase, appears when the system is parametrically driven. Employing different methods, we characterize the different phases and the transitions between them. Specific heed is paid to the role of dissipation in determining the phase boundaries. Our analysis paves the road for the experimental study of dynamically stabilized phases of interacting light and matter.
Abstract Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.
This study formulates observer-dependent Krylov spreading for non-inertial quantum systems driven by linear Bogoliubov transformations. Starting with the closed single Rindler-pair SU(1,1)SU(1,1)SU(1,1) sector, we show that its Lanczos basis is identical to the Rindler pair-number basis. As a result, the Krylov spread complexity reduces exactly to the mean number of correlated Rindler pairs, Ck=∣βk∣2C_k=\vertβ_k\vert^2Ck​=∣βk​∣2. Within this framework, we demonstrate that Krylov spreading dynamics are governed by the competition between the detuning parameter and the coupling constant, naturally dividing the dynamics into three distinct regimes. Notably, Krylov complexity becomes localized in the detuning-dominated regime. By extending this to a multimode, strictly quadratic Bogoliubov Hamiltonian, we find that inequivalent Rindler wave-packet pairs violate the Ck=∣βk∣2C_k=\vertβ_k\vert^2Ck​=∣βk​∣2 correspondence, thereby highlighting the single-pair SU(1,1)SU(1,1)SU(1,1) model as an exactly solvable, observer-adapted benchmark. In such multimode scenarios, the mean pair-number eigenstates no longer dictate Krylov complexity. Overall, our work provides a new perspective for analyzing Krylov complexity in non-inertial quantum systems.
The Dicke model is a staple of theoretical cavity quantum electrodynamics (cavity QED), describing the interaction between an ensemble of atoms and a single radiation mode of an optical cavity. It is studied both quantum mechanically and semiclassically for two-level atoms, and demonstrates a rich variety of dynamics such as phase transitions, phase multistability, and chaos. In this work we explore an open, spin-1 Dicke model with independent co and counterrotating coupling terms as well as a quadratic Zeeman shift enabling control over the atomic energy-level structure. We investigate the stability of operator and moment equations under two approximations and show the system undergoes phase transitions. To compliment these results, we relax the aforementioned approximations and investigate the system semiclassically. We show evidence of phase transitions to steady-state and oscillatory superradiance in this semiclassical model, as well as the emergence of chaotic dynamics. The varied and complex behaviors admitted by the model highlights the need to more rigorously map its dynamics.
Systems of strongly interacting atoms and photons, which can be realized wiring up individual cavity QED systems into lattices, are perceived as a new platform for quantum simulation. While sharing important properties with other systems of interacting quantum particles, here we argue that the nature of light-matter interaction gives rise to unique features with no analogs in condensed matter or atomic physics setups. By discussing the physics of a lattice model of delocalized photons coupled locally with two-level systems through the elementary light-matter interaction described by the Rabi model, we argue that the inclusion of counterrotating terms, so far neglected, is crucial to stabilize finite-density quantum phases of correlated photons out of the vacuum, with no need for an artificially engineered chemical potential. We show that the competition between photon delocalization and Rabi nonlinearity drives the system across a novel Z(2) parity symmetry-breaking quantum criticality between two gapped phases that share similarities with the Dicke transition of quantum optics and the Ising critical point of quantum magnetism. We discuss the phase diagram as well as the low-energy excitation spectrum and present analytic estimates for critical quantities.
Quantum many-body systems driven far from equilibrium can exhibit phase transitions, chaos and entanglement, but probing these phenomena in large, nearly closed systems remains challenging. The Dicke model, which couples an ensemble of two-level systems collectively to a single bosonic mode, provides a fundamental framework for exploring these phenomena in light-matter interactions. Here we show that a two-dimensional crystal of approximately 100 trapped ions realizes this model and accesses integrable dynamical phase transitions, signatures of chaotic dynamics and fluctuation-driven spin-phonon excitation growth. Adiabatic elimination of the vibrational mode reveals ferromagnetic and paramagnetic dynamical spin phases, whereas strong spin-phonon coupling produces irregular trajectories and local relaxation consistent with quantum thermalization. Following a quench from an unstable fixed point, correlated excitation growth precedes generalized vacuum Rabi collapses and revivals; calibrated calculations identify two-mode squeezing 2.6 dB below the standard quantum limit, supported by reconstruction from spin measurements. By connecting collective dynamical instabilities to experimentally accessible signatures of quantum correlations, these results establish large ion crystals as a scalable platform for investigating non-equilibrium light-matter dynamics and generating resources for quantum-enhanced sensing.