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Optica Advanced Photonics Congress

From Quantum to Classical Nonlinear Dynamics in the Unbalanced Dicke Model

Kevin C. Stitely, Andrus Giraldo, Bernd Krauskopf, A. S. Parkins

Published Jan 1, 2022• 1 citation
Publisher

Abstract

We investigate the appearance of classical nonlinear dynamics in the non-equilibrium quantum phase transition to superradiance in the unbalanced Dicke model. Specifically, we uncover a variety of dynamical behavior, including hysteresis, superradiant oscillations, and chaos.
DOI
10.1364/np.2022.npw2g.2

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In 1973 Klaus Hepp and Elliott Lieb showed that a tightly confined ensemble of atoms placed in a quantised single-mode optical cavity will have their emission of light enhanced by constructive interference of emission events if the photons emitted are indistinguishable. This effect, known as superradiance, has proven difficult to realise experimentally due to the strong light-matter coupling required to produce such behaviour. However, over the last decade there have been several successful realisations which engineer superradiance by driving Raman transitions between atomic ground states of multilevel atoms. In this case, superradiance emerges via a quantum phase transition mediated by the laser driving power. In this work, we consider a generalisation of this model referred to as the unbalanced Dicke model. It has recently been implemented experimentally and is characterised by two lasers driving two distinct Raman transitions. We show that, in the semiclassical limit of a large number of atoms, the unbalanced Dicke model exhibits a wealth of nonlinear dynamics which describe novel dynamical phases of matter. Examples of nonlinear behaviour observed are pitchfork, saddle-node, and Hopf bifurcations, as well as two types of chaos. We explore the quantum mechanical manifestations of these bifurcations and additional physical effects which appear in the presence of quantum fluctuations. In this way, the model creates a playground in which to explore the onset of classical nonlinear dynamics from an open quantum system.
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 that 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 corotating 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 complex dynamics.Examples of the 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 road map 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, thus, providing the possibility of studying a manifestation of quantum chaos in a specific, experimentally realizable system.
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 counterrotating 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 is observed clearly in quantum simulations.
In this thesis, we explore the Dicke model, well-known for its explanation of the superradiant phase transition and signatures of Classical and Quantum Chaos. We then propose a variational approach to determine the semi-classical (SC) equations of motion, using coherent states. The SC dynamics exhibits signs of classical chaos both with increasing coupling and as the energy of the system is gradually increased, eventually leading to it completely engulfing the 2 dimensional phase space of the pseudospin, which we examine through Poincare sections. We then use numerical diagonalization to study the Quantum Dicke model - it is observed that at and after exactly the critical value of the coupling, the system shows a net macroscopic photon expectation value, and the pseudospin component in the z direction starts to steadily decline - implying a net fraction of the atoms are shifting to the excited state. We then study the dynamics of the Quantum Dicke Hamiltonian using coherent states and compare the expectation values of the dimensional with the classical trajectories from the semi-classical dynamics for stable and chaotic orbits. We then introduce a linear quench which drives the Dicke model across the Superradiant phase transition by changing the coupling strength. The quenched dynamics is simulated using both semi-classical and quantum dynamics - it is seen that dynamically varying quantities are periodically oscillating around their instantaneous steady-state coupling values.
We study the nonlinear, semiclassical dynamics of an open spin-1 (three-level) variant of the traditional Dicke model. In particular, we focus on V-type energy-level configurations with varying degrees of energy-level asymmetry. We also allow for unbalanced coupling -- where co-rotating and counter-rotating Hamiltonian terms are independently tunable. We characterise the system with a dynamical systems approach, where different behaviors map to definite dynamical objects, and phase transitions to bifurcations. We find the emergence of both periodic and two-frequency oscillations, as well as multistability and chaotic dynamics.