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arXiv:2004.04486 · Physical Review Research

Nonlinear semiclassical dynamics of the unbalanced, open Dicke model

Kevin C. Stitely, Andrus Giraldo, Bernd Krauskopf, Scott Parkins

Published Jul 23, 2020• 43 citations
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Abstract

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.

DOI
10.1103/physrevresearch.2.033131

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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 study a generalization of a recently introduced Dicke trimer model [Phys. Rev. Lett. 128, 163601 (2022); Phys. Rev. Res. 5, L042016 (2023)], which allows for cavity losses and unbalanced light-matter interactions (in which rotating and counter-rotating terms can be tuned independently). In the original description of a Dicke trimer, three Dicke models are coupled in a ring topology via a complex photon hopping whose complex phase describes a synthetic magnetic field threading the loop. This original model features several intriguing equilibrium phases and critical phenomena such as frustrated superradiance, two-critical scalings in the frustrated superradiant phase, and finite critical fluctuations in the anomalous normal phase. Here, we find that in the extreme unbalanced limit, where only rotating terms are present, the U(1) symmetry of the Tavis-Cummings model is restored, qualitatively altering the critical phenomena in the superradiant phase due to the presence of a zero-energy mode. To analyze this general regime, we develop a semiclassical theory based on a requantization technique. This theory also provides further physical insight on recently reported anomalous finite critical fluctuations in the time-reversal broken regime. Moving to the open-Dicke case, by introducing local dissipation to the cavities, we observe the emergence of a rich range of nonequilibrium phases characterized by trivial and nontrivial dynamical signatures. In the former case, we identify, when time-reversal symmetry is present, a new stationary phase that features superradiant states in two of the three cavities and a normal state in the other cavity. In the latter case, we observe the emergence of dynamical phases in which the system exhibits superradiant oscillations, characterized by periodic or chaotic phase space patterns. The landscape of transitions associated with these dynamical phases features a wide range of qualitatively different behaviors such as Hopf bifurcations (followed by period-doubling cascades or quasiperiodic oscillations), anomalous Hopf bifurcations (with burst-oscillation-like post-bifurcation dynamics), collisions between basins of attraction (associated with different symmetry-broken equilibria), and exterior crises (featuring transient chaotic dynamics). We highlight how the two-critical-scalings feature of the closed model is robust under dissipation (with doubled critical exponents) while the phenomenon of anomalous finite critical fluctuations becomes a mean-field scaling (as a consequence of Hopf bifurcations of the equilibria featuring the normal state) in the open model.
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.
We study a generalization of a recently introduced Dicke trimer model [Phys. Rev. Lett. 128, 163601, Phys. Rev. Research 5, L042016], which allows for cavity losses and unbalanced light-matter interactions (in which rotating and counter-rotating terms can be tuned independently). We find that in the extreme unbalanced limit, the U(1)U(1)U(1) symmetry of the Tavis-Cummings model is restored, qualitatively altering the critical phenomena in the superradiant phase due to the presence of a zero-energy mode. To analyze this general regime, we develop a semiclassical theory based on a re-quantization technique. This theory also provides further physical insight on a recently reported anomalous finite critical fluctuations in the time-reversal broken regime. Moving to the open-Dicke case, by introducing local dissipation to the cavities, we observe the emergence of a rich range of nonequilibrium phases characterized by trivial and non-trivial dynamical signatures. In the former case, we identify, when time-reversal symmetry is present, a new stationary phase that features superradiant states in two of the three cavities and a normal state in the other cavity. In the latter case, we observe the emergence of dynamical phases in which the system exhibits superradiant oscillations, characterized by periodic or chaotic phase space patterns. The landscape of transitions associated with these dynamical phases features a wide range of qualitatively different behaviours such as Hopf bifurcations, anomalous Hopf bifurcations, collisions between basins of attraction, and exterior crises. We highlight how the two-critical-scalings feature of the closed model is robust under dissipation while the phenomenon of anomalous finite critical fluctuations becomes a mean-field scaling in the open model.
We study the effect of spontaneous emission and incoherent atomic pumping on the nonlinear semiclassical dynamics of the unbalanced Dicke model---a generalization of the Dicke model that features independent coupling strengths for the co- and counter-rotating interaction terms. As well as the ubiquitous superradiant behavior the Dicke model is well-known for, the addition of spontaneous emission combined with the presence of strong counter-rotating terms creates a laserlike behavior termed counter-lasing. These states appear in the semiclassical model as stable periodic orbits. We perform a comprehensive dynamical analysis of the appearance of counter-lasing in the unbalanced Dicke model subject to strong cavity dissipation, such that the cavity field can be adiabatically eliminated to yield an effective Lipkin-Meshkov-Glick (LMG) model. If the coupling strength of the corotating interactions is small, then the counter-lasing phase appears via a Hopf bifurcation of the deexcited state. We find that if the rate of spontaneous emission is small, this can lead to resurgent superradiant pulses. However, if the corotating coupling is larger, then the counter-lasing phase must emerge via the steady-state superradiant phase. Such a transition is the result of the competition of the coherent and incoherent processes that drive superradiance and counter-lasing, respectively. We observe a surprisingly complex transition between the two, associated with the formation of a chaotic attractor over a thin transitional parameter region.
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.
We study the complex nonlinear dynamics of the two-photon Dicke model in the semiclassical limit by considering cavity and qubit dissipation. In addition to the normal and super-radiant phases, another phase that contains abundant chaos-related phenomena is found under balanced rotating and counter-rotating couplings. In particular, chaos may manifest itself through period-doubling bifurcation, intermittent chaos, or quasi-periodic oscillation, depending on the value of qubit frequency. Transition mechanisms that exist in these three distinct routes are investigated through the system's long-time evolution and bifurcation diagram. Additionally, we provide a comprehensive phase diagram detailing both the existence of stable fixed points and the aforementioned chaos-related dynamics.