Open Dicke Model Research Articles

Discrete time crystal in an open optomechanical system

The spontaneous breaking of time translation symmetry in periodically driven Floquet systems can lead to a discrete time crystal. Here we study the occurrence of such dynamical phase in a driven-dissipative optomechanical system with two membranes in the middle. We find that, under certain conditions, the system can be mapped to an open Dicke model and realizes a superradiant-type phase transition. Furthermore, applying a suitable periodically modulated drive, the system dynamics exhibits a robust subharmonic oscillation persistent in the thermodynamic limit. Published by the American Physical Society 2024

Analysis of chaos and regularity in the open Dicke model.

We present an analysis of chaos and regularity in the open Dicke model, when dissipation is due to cavity losses. Due to the infinite Liouville space of this model, we also introduce a criterion to numerically find a complex spectrum which approximately represents the system spectrum. The isolated Dicke model has a well-defined classical limit with two degrees of freedom. We select two case studies where the classical isolated system shows regularity and where chaos appears. To characterize the open system as regular or chaotic, we study regions of the complex spectrum taking windows over the absolute value of its eigenvalues. Our results for this infinite-dimensional system agree with the Grobe-Haake-Sommers (GHS) conjecture for Markovian dissipative open quantum systems, finding the expected 2D Poisson distribution for regular regimes, and the distribution of the Ginibre unitary ensemble (GinUE) for the chaotic ones, respectively.

Robust entanglement and steering in open Dicke models with individual atomic spontaneous emission and dephasing.

In this paper, we study steady-state quantum entanglement and steering in an open Dicke model where cavity dissipation and individual atomic decoherence are taken into account. Specifically, we consider that each atom is coupled to independent dephasing and squeezed environments, which makes the widely-adopted Holstein-Primakoff approximation invalid. By discovering the features of quantum phase transition in the presence of the decohering environments, we mainly find that (i) in both normal and superradiant phases, the cavity dissipation and individual atomic decoherence can improve the entanglement and steering between the cavity field and atomic ensemble; (ii) the individual atomic spontaneous emission leads to the appearance of the steering between the cavity field and atomic ensemble but the steering in two directions cannot be simultaneously generated; (iii) the maximal achievable steering in normal phase is stronger than that in superradiant phase; (iv) the entanglement and steering between the cavity output field and the atomic ensemble are much stronger than that with the intracavity, and the steerings in two directions can be achieved even with the same parameters. Our findings reveal unique features of quantum correlations in the open Dicke model in the presence of individual atomic decoherence processes.

Critical behavior of quantum Fisher information in finite-size open Dicke model

We explore the steady-state critical behavior of the finite-size open Dicke model—a model that incorporates spontaneous emission decay of the collective atomic spin states and decay of the cavity field. From the perspective of quantum information theory, we can often better characterize the quantum phase transition. In this paper, we characterize the super-radiant phase transition of the steady state of the open Dicke model by numerically calculating the quantum Fisher information (QFI). We calculate the QFI for the atomic state and the cavity field state, as well as their derivatives. We find that the QFI of the cavity field state is more sensitive to atomic decay, and is suppressed more severely in the presence of atomic decay. In contrast, the QFI of the atomic state is less sensitive to the photon loss of the cavity field.

Driven-dissipative Ising model: Dynamical crossover at weak dissipation

Driven quantum systems coupled to an environment typically exhibit effectively thermal behavior with relaxational dynamics near criticality. However, a different qualitative behavior might be expected in the weakly dissipative limit due to the competition between coherent dynamics and weak dissipation. In this work, we investigate a driven-dissipative infinite-range Ising model in the presence of individual atomic dissipation, a model that emerges from the paradigmatic open Dicke model in the large-detuning limit. We show that the system undergoes a dynamical crossover from relaxational dynamics, with a characteristic dynamical exponent , to underdamped critical dynamics governed by the exponent in the weakly dissipative regime; a behavior that is markedly distinct from that of equilibrium. Finally, utilizing an exact diagrammatic representation, we demonstrate that the dynamical crossover to underdamped criticality is not an artifact of the mean-field nature of the model and persists even in the presence of short-range perturbations.

Driven-dissipative Ising model: An exact field-theoretical analysis

Driven-dissipative many-body systems are difficult to analyze analytically due to their non-equilibrium dynamics, dissipation and many-body interactions. In this paper, we consider a driven-dissipative infinite-range Ising model with local spontaneous emission, which naturally emerges from the open Dicke model in the large-detuning limit. Utilizing an adaptation of the Suzuki-Trotter quantum-to-classical mapping, we develop an exact field-theoretical analysis and a diagrammatic representation of the spin model that can be understood from a simple scattering picture. With this representation, we are able to analyze critical behavior, finite-size scaling and the effective temperature near the respective phase transition. Our formalism further allows a detailed study of the ordered phase where we find a "heating" region within which the effective temperature becomes negative, thereby exhibiting a truly non-equilibrium behavior. At the phase transition, we find two distinct critical behaviors with overdamped and underdamped critical dynamics at generic and weakly-dissipative critical points, respectively. We further show that the underdamped critical behavior is robust against short-range perturbations and is not an artifact of the mean-field nature of the model. To treat such perturbations, we extend our diagrammatic representation to include the coupling to spin waves due to the short-range interactions. The field-theoretical approach and the diagrammatics developed in this work should prove useful in applications to generic short-range driven-dissipative spin systems.

Superradiant switching, quantum hysteresis, and oscillations in a generalized Dicke model

We demonstrate quantum signatures of deterministic nonlinear dynamics in the transition to superradiance of a generalized open Dicke model with different coupling strengths for the co- and counter-rotating light-matter interaction terms. A first-order phase transition to coexisting normal and superradiant phases is observed, corresponding with the emergence of switching dynamics between these two phases, driven by quantum fluctuations. We show that this phase coexistence gives rise to a hysteresis loop also for the quantum mechanical system. Additionally, a transition to a superradiant oscillatory phase can be observed clearly in quantum simulations.

On the Open Dicke-Type Model Generated by an Infinite-Component Vector Spin

We consider an open Dicke model comprising a single infinite-component vector spin and a single-mode harmonic oscillator which are connected by Jaynes–Cummings-type interaction between them. This open quantum model is referred to as the OISD (Open Infinite-component Spin Dicke) model. The algebraic structure of the OISD Liouvillian is studied in terms of superoperators acting on the space of density matrices. An explicit invertible superoperator (precisely, a completely positive trace-preserving map) is obtained that transforms the OISD Liouvillian into a sum of two independent Liouvillians, one generated by a dressed spin only, the other generated by a dressed harmonic oscillator only. The time evolution generated by the OISD Liouvillian is shown to be asymptotically equivalent to that generated by an adjusted decoupled Liouvillian with some synchronized frequencies of the spin and the harmonic oscillator. This asymptotic equivalence implies that the time evolution of the OISD model dissipates completely in the presence of any (tiny) dissipation.

Nonlinear semiclassical dynamics of the unbalanced, open Dicke model

In recent years there have been significant advances in the study of many-body interactions between atoms and light confined to optical cavities. One model which has received widespread attention of late is the Dicke model, which under certain conditions exhibits a quantum phase transition to a state in which the atoms collectively emit light into the cavity mode, known as superradiance. We consider a generalization of this model that features independently controllable strengths of the co- and counter-rotating terms of the interaction Hamiltonian. We study this system in the semiclassical (mean field) limit, i.e., neglecting the role of quantum fluctuations. Under this approximation, the model is described by a set of nonlinear differential equations, which determine the system's semiclassical evolution. By taking a dynamical systems approach, we perform a comprehensive analysis of these equations to reveal an abundance of novel and complex dynamics. Examples of the novel phenomena that we observe are the emergence of superradiant oscillations arising due to Hopf bifurcations, and the appearance of a pair of chaotic attractors arising from period-doubling cascades, followed by their collision to form a single, larger chaotic attractor via a sequence of infinitely many homoclinic bifurcations. Moreover, we find that a flip of the collective spin can result in the sudden emergence of chaotic dynamics. Overall, we provide a comprehensive roadmap of the possible dynamics that arise in the unbalanced, open Dicke model in the form of a phase diagram in the plane of the two interaction strengths. Hence, we lay out the foundations to make further advances in the study of the fingerprint of semiclassical chaos when considering the master equation of the unbalanced Dicke model, that is, the possibility of studying a manifestation of quantum chaos in a specific, experimentally realizable system.

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems.

Discrete time crystals are a recently proposed and experimentally observed out-of-equilibrium dynamical phase of Floquet systems, where the stroboscopic dynamics of a local observable repeats itself at an integer multiple of the driving period. We address this issue in a driven-dissipative setup, focusing on the modulated open Dicke model, which can be implemented by cavity or circuit QED systems. In the thermodynamic limit, we employ semiclassical approaches and find rich dynamical phases on top of the discrete time-crystalline order. In a deep quantum regime with few qubits, we find clear signatures of a transient discrete time-crystalline behavior, which is absent in the isolated counterpart. We establish a phenomenology of dissipative discrete time crystals by generalizing the Landau theory of phase transitions to Floquet open systems.

Superradiant to subradiant phase transition in the open system Dicke model: dark state cascades

Collectivity in ensembles of atoms gives rise to effects like super- and subradiance. While superradiance is well studied and experimentally accessible, subradiance remains elusive since it is difficult to track experimentally as well as theoretically. Here we present a new type of phase transition in the resonantly driven, open Dicke model that leads to a deterministic generation of subradiant states. At the transition the system switches from a predominantly superradiant to a predominantly subradiant state. Counterintuitively, the cavity decay is the crucial parameter for subradiant state generation and not the individualizing process of spontaneous decay. The observed effect is thus a cavity assisted generation of subradiant quantum coherences. Clear experimental signatures for the effect are presented and entanglement properties are discussed. Letting the system relax into the ground state generates a cascade of dark Dicke states, with dark state populations up to unity. Furthermore we introduce a collectivity measure that allows to quantify collective behaviour.

Many-body quantum optics with decaying atomic spin states: ( γ, κ ) Dicke model

We provide a theory for quantum-optical realizations of the open Dicke model with internal, atomic spin states subject to spontaneous emission with rate $\gamma$. This introduces a second decay channel for excitations to irreversibly dissipate into the environment, in addition to the photon loss with rate $\kappa$, which is composed of individual atomic decay processes and a collective atomic decay mechanism. The strength of the latter is determined by the cavity geometry. We compute the mean-field non-equilibrium steady states for spin and photon observables in the long-time limit, $t\rightarrow \infty$. Although $\gamma$ does not conserve the total angular momentum of the spin array, we argue that our solution is exact in the thermodynamic limit, for the number of atoms $N\rightarrow \infty$. In light of recent and upcoming experiments realizing superradiant phase transitions using internal atomic states with pinned atoms in optical lattices, our work lays the foundation for the pursuit of a new class of open quantum magnets coupled to quantum light.

Critical relaxation with overdamped quasiparticles in open quantum systems

We study the late-time relaxation following a quench in a driven-dissipative quantum many-body system. We consider the open Dicke model, describing the infinite-range interactions between $N$ atoms and a single, lossy electromagnetic mode. We show that the dynamical phase transition at a critical atom-light coupling is characterized by the interplay between reservoir-driven and intrinsic relaxation processes. Above the critical coupling, small fluctuations in the occupation of the dominant quasiparticle-mode start to grow in time while the quasiparticle lifetime remains finite due to losses. Near the critical interaction strength we observe a crossover between exponential and power-law $1/\tau$ relaxation, the latter driven by collisions between quasiparticles. For a quench exactly to the critical coupling, the power-law relaxation extends to infinite times, but the finite lifetime of quasiparticles prevents ageing to appear. We predict our results to be accessible to quench experiments with ultracold bosons in optical resonators.

Dynamical phase transition in the open Dicke model

The Dicke model with a weak dissipation channel is realized by coupling a Bose-Einstein condensate to an optical cavity with ultranarrow bandwidth. We explore the dynamical critical properties of the Hepp-Lieb-Dicke phase transition by performing quenches across the phase boundary. We observe hysteresis in the transition between a homogeneous phase and a self-organized collective phase with an enclosed loop area showing power-law scaling with respect to the quench time, which suggests an interpretation within a general framework introduced by Kibble and Zurek. The observed hysteretic dynamics is well reproduced by numerically solving the mean-field equation derived from a generalized Dicke Hamiltonian. Our work promotes the understanding of nonequilibrium physics in open many-body systems with infinite range interactions.

Cavity-Mediated Near-Critical Dissipative Dynamics of a Driven Condensate

We investigate the near-critical dynamics of atomic density fluctuations in the nonequilibrium self-organization transition of an optically driven quantum gas coupled to a single mode of a cavity. In this system cavity-mediated long-range interactions between atoms, tunable by the drive strength, lead to softening of an excitation mode recently observed in experiments. This phenomenon has previously been studied within a two-mode approximation for the collective motional degrees of freedom of the atomic condensate, which results in an effective open-system Dicke model. Here, including the full spectrum of atomic modes we find a finite lifetime for a rotonlike mode in the Bogoliubov excitation spectrum that is strongly pump dependent. The corresponding decay rate and critical exponents for the phase transition are calculated explaining the nonmonotonic pump-dependent atomic damping rate observed in recent experiments. We compute the near-critical behavior of the intracavity field fluctuations that has been previously shown to be enhanced with respect to the equilibrium Dicke model in a two-mode approximation. We highlight the role of the finite size of the system in the suppression of it below the expectations of the open Dicke model.