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arXiv:1507.01811 · Physical Review A

Dissipative two-mode Tavis-Cummings model with time-delayed feedback control

Wassilij Kopylov, Milan Radonjić, Tobias Brandes, Antun Balaž, Axel Pelster

Published Dec 21, 2015• 28 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.92.063832

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Journal of the Optical Society of America B•Jul 1, 1999•1 citation
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.
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.
Abstract We investigate the steady-state quantum phases and associated collective phenomena of an open Tavis–Cummings (TC) model driven by a two-photon source. The standard (non-dissipative) TC model describes the quantum phases of an ensemble of N q two-level quantum emitters interacting with a single-mode electromagnetic field. The interplay among coherent driving, dissipation, and dipole interactions of the open TC model results in emergent collective phenomena, leading to a dissipative or non-equilibrium phase transition from the normal to the superradiant phase. We solve the Liouvillian equation analytically using the semi-classical mean-field approximation. We carry out stability analysis of the steady-state phases and determine the phase boundary. The use of Holstein–Primakoff transformation in the thermodynamic limit N q → ∞ reduces the system to an effective coupled-oscillator model, the solution of which yields collective modes.
We propose a time-delayed feedback control scheme for open quantum systems that can dramatically reduce the time to reach steady state. No measurement is performed in the feedback loop, and we suggest a simple all-optical implementation for a cavity QED system. We demonstrate the potential of the scheme by applying it to a driven and dissipative Dicke model, as recently realized in a quantum gas experiment. The time to reach steady state can be reduced by two orders of magnitude for the parameters taken from the experiment, making previously inaccessible long time attractors reachable within typical experimental run times. The scheme also offers the possibility of slowing down the dynamics, as well as qualitatively changing the phase diagram of the system.
This thesis discusses the dynamics of quantum-optical systems located in complex, non-markovian photonic environments, using numerical simulations as well as analytical derivations. The main focus lies on time-delayed quantum-coherent feedback effects of the Pyragas type, in analogy to effects well-known in classical physics. Time-delayed feedback is treated as a specific kind of a non-markovian environment. Additionally, the control of qubit entanglement in optical cavities is studied. The thesis consists of three main parts: After an overview of the important concepts of quantum optics, advanced methods for the description and numerical treatment of time-delayed quantum-coherent feedback are presented. The main focus lies on the development of a pseudomode-based approach, which describes time-delayed feedback as the coupling to a complex network of lossy harmonic oscillators. Additionally, further methods for numerical simplification are presented, in particular nonlinear expectation value dynamics for the treatment of arbitrary non-markovian reservoirs, as well as the use of input-output theory for time-delayed feedback. In the second part, applications of time-delayed feedback to control quantum statistics are discussed. First it is demonstrated that time-delayed feedback can be used to create and sustain entanglement between coupled qubits. This approach will then be extended to larger networks of qubits. Afterwards, feedback control is applied to nonlinear quantum-optical systems: It is shown how to use it for the stabilization of Fock states in a cavity containing a Kerr medium, and this analysis is expanded to a cavity containing a two-level system. Furthermore, it is shown how time-delayed feedback can be used to control and enhance the entanglement of photons emitted in a biexciton cascade. In the third and last part, the non-equilibrium dynamics of qubits coupled to a photonic environment consisting of high-Q cavities subject to periodic driving are discussed. First, the connection between bistability and entanglement in a cavity containing two qubits is analyzed. Next a setup of two coupled cavities, containing a qubit each, is examined. The influence of a third cavity in between the two other cavities, simulating a delay line, is analyzed. Furthermore, a protocol is developed on how to overcome dephasing between the two qubits, using a resonant Raman scattering process. It is shown that this protocol can create and sustain high values of entanglement, and is ready to be extended to systems of more than two qubits.
We study theoretically the dynamic behavior of a J′=1→J″=0 laser (J is the angular momentum quantum number) optically pumped by means of a linearly polarized coherent field (coupled to an adjacent transition J=0→J′=1) when the pump field polarization is rotated at a constant angular velocity Ω and the laser field polarization either is fixed by the cavity or is free (isotropic cavity). Because of a strong pump-induced gain anisotropy, the dynamic behavior is completely different in each case. In the case of fixed laser field polarization, rich amplitude dynamics, which depend on the pump field strength, are found. At slow modulation frequencies (with respect to the molecular and the cavity relaxation rates), the system does not always follow the sequence of stationary and dynamic solutions that correspond to the autonomous laser as a function of the relative orientation angle between the polarizations of the pump and the laser fields. Phenomena such as delayed switching and suppression of chaos and stabilization of unstable steady states are found. In the case of an isotropic cavity, the pump field vector rotation is transferred to the laser field vector and amplitude unstable regimes are also strongly inhibited.