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arXiv:1403.0620 · New Journal of Physics

Time-delayed feedback control of the Dicke–Hepp–Lieb superradiant quantum phase transition

Wassilij Kopylov, Clive Emary, Eckehard Schöll, Tobias Brandes

Published Jan 20, 2015• 30 citations
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Abstract

We apply the time-delayed Pyragas control scheme to the dissipative Dicke model via a modulation of the atom-field-coupling. The feedback creates an infinite sequence of non-equilibrium phases with fixed points and limit cycles in the primary superradiant regime. We analyse this Hopf bifurcation scenario as a function of delay time and feedback strength and determine analytical conditions for the phase boundaries.

DOI
10.1088/1367-2630/17/1/013040

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quant-pharXiv:2604.18119•Apr 21, 2026•0 citations
We apply the time-delayed Pyragas control scheme to the dissipative Dicke model via a modulation of the atom-field-coupling. The feedback creates an infinite sequence of non-equilibrium phases with fixed points and limit cycles in the primary superradiant regime. We analyse this Hopf bifurcation scenario as a function of delay time and feedback strength and determine analytical conditions for the phase boundaries.
In this theoretical thesis we apply the time-delayed Pyragas feedback scheme to several many-body quantum systems in order to control the quantum phases in a rather new way. The systems are the dissipative Dicke model, the dissipative Lipkin-Meshkov-Glick (LMG) model and a two-mode Tavis-Cummings laser. In the dissipative Dicke model our chosen way of feedback control, namely the modulation of the atom-field coupling by a difference of emitted photon numbers at different times, creates additional new quantum phases. These show up as an infinite sequence of super- and subcritical Hopf bifurcations of the stationary state by generation of stable and unstable limit cycles. We analyze and discuss the appearing bifurcations as a function of time delay and feedback strength and calculate an analytical expression for the phase boundaries. On top of this, we argue that our scheme is easy to implement within existing experimental setups, as in experiments the coupling is tuned by laser intensity. In the dissipative LMG model we investigate how the dissipation affects the excited state quantum phase transition (ESQPT). We show that with dissipative effects the ESQPT is directly visible in the spectrum of the effective Hamiltonian and not only in the density of states. Moreover, the ESQPT signal in the system observables gets smoothed. Then we use time delayed feedback control to restore the ESQPT signal by creation of new phases. We analyze and discuss the behavior of the system for different time delays. We argue that using our feedback scheme it is easier to measure ESQPT in the open LMG system than without it in the closed one. Then we use a 2-mode laser based on the Tavis-Cummings model with additional incoherent pumping and decay channels. Instead of the usually used rate-equation model, we describe the two-mode laser starting from a quantum mechanical description and present an analytical solution of the corresponding stationary mean-field equations in thermodynamic limit. We analyze the stationary solutions and their stability and obtain a complex phase diagram with up to 4 fixed points. In addition to this, we apply different time delayed Pyragas feedback control schemes, which allow us to influence the laser dynamic. The frequency of the emitted laser light can be controlled then or an unstable fixed point stabilized. In this way, our thesis links the topics of control, phase transitions and dissipation together which induce an interesting and unexpected dynamics in quantum many-body systems. Although all final calculations are at the mean-field level, we hold them to be valid in thermodynamic limit.
We investigate the dynamics of a two-mode laser system by extending the two-mode Tavis-Cummings model with dissipative channels and incoherent pumping and by applying the mean-field approximation in the thermodynamic limit. To this end we analytically calculate up to four possible nonequilibrium steady states (fixed points) and determine the corresponding complex phase diagram. Various possible phases are distinguished by the actual number of fixed points and their stability. In addition, we apply three time-delayed Pyragas feedback control schemes. Depending on the time delay and the strength of the control term, this can lead to the stabilization of unstable fixed points or to the selection of a particular cavity mode that is macroscopically occupied.
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.
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 present the non-equilibrium phase diagram of a model which can demonstrate both Dicke–Hepp–Lieb superradiance and regular lasing by varying the coherent and incoherent driving terms We find that the regions in the phase diagram corresponding to superradiance and standard lasing are always separated by a normal region. We analyse the behaviour of the system using a combination of exact numerics based on permutation symmetry of the density matrix for small to intermediate numbers of molecules, and second order cumulant equations for large numbers of molecules. We find that the nature of the photon distribution in the superradiant and lasing states are very similar, but the emission spectrum is very different. We also show that in the presence of both coherent and incoherent driving, a period-doubling route to a chaotic state occurs.
The interplay between many-body interactions and controlled dissipation provides a rich setting for nonequilibrium quantum phases. We explore the superradiant phase transitions in a Rydberg-interacting open Dicke model with cavity loss and local dissipation. We find that the nonlinear feedback from atomic interactions stabilizes the atomic collective excited state and induces resonant collective motion, which can be further converted into robust superradiant oscillations with the inclusion of local dissipation. Beyond the mean-field approximation, we prove this system can support the finite-size local dissipative temporal order, i.e., continuous time crystal, from evidences of the decreasing damping of a finite-frequency Liouvillian mode and the linear-size scaling of quantum fluctuations. The correlation between the atomic colletive dynamics and the cavity photon number makes our findings easy to detect from output optical signal. By bridging interacting atomic matter and driven-dissipative cavity QED, our Rydberg-interacting Dicke model provides a unique spin-1/2 platform for exploring dynamical phase transitions and macroscopic temporal order in open quantum matter.