Pyragas control allows us to stabilize unstable states in applied nonlinear science. We propose to apply a quantum version of the Pyragas protocol to control individual photon probabilities in an otherwise only globally accessible photon-probability distribution of a quantum light emitter. The versatility of quantum Pyragas control is demonstrated for the case of a two-level emitter in a pulsed laser-driven half cavity. We show that one- and two-photon events respond in a qualitatively different way to the half-cavity-induced feedback signal. One-photon events are either enhanced or suppressed, depending on the choice of parameters. In contrast, two-photon events undergo exclusively an enhancement up to $50%$ for the chosen pulse areas. We hereby propose an implementation of quantum Pyragas control via a time-delayed feedback setup.
Pyragas control allows us to stabilize unstable states in applied nonlinear science. We propose to apply a quantum version of the Pyragas protocol to control individual photon probabilities in an otherwise only globally accessible photon-probability distribution of a quantum light emitter. The versatility of quantum Pyragas control is demonstrated for the case of a two-level emitter in a pulsed laser-driven half cavity. We show that one- and two-photon events respond in a qualitatively different way to the half-cavity-induced feedback signal. One-photon events are either enhanced or suppressed, depending on the choice of parameters. In contrast, two-photon events undergo exclusively an enhancement up to 50 for the chosen pulse areas. We hereby propose an implementation of quantum Pyragas control via a time-delayed feedback setup.
Quantum coherent feedback control is a measurement-free control method fully preserving quantum coherence. In this paper we show how time-delayed quantum coherent feedback can be used to control the degree of squeezing in the output field of a cavity containing a degenerate parametric oscillator. We focus on the specific situation of Pyragas-type feedback control where time-delayed signals are fed back directly into the quantum system. Our results show how time-delayed feedback can enhance or decrease the degree of squeezing as a function of time delay and feedback strength.
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 present a Heisenberg-operator-based formulation of coherent quantum feedback and Pyragas control. This model is easy to implement and allows for an efficient and fast calculation of the dynamics of feedback-driven observables as the number of contributing correlations grows in systems with a fixed number of excitations only linearly in time. Furthermore, our model unravels the quantum kinetics of entanglement growth in the system by explicitly calculating non-Markovian multi-time correlations, e.g., how the emission of a photon is correlated with an absorption process in the past. Therefore, the time-delayed differential equations are expressed in terms of insightful physical quantities. Another considerable advantage of this method is its compatibility with typical approximation schemes, such as factorization techniques and the semi-classical treatment of coherent fields. This allows the application on a variety of setups, ranging from closed quantum systems in the few excitation regimes to open systems and Pyragas control in general.
We propose a scheme to control cavity quantum electrodynamics in the single photon limit by delayed feedback. In our approach a single emitter-cavity system, operating in the weak coupling limit, can be driven into the strong coupling-type regime by an external mirror: The external loop produces Rabi oscillations directly connected to the electron-photon coupling strength. As an expansion of typical cavity quantum electrodynamics, we treat the quantum correlation of external and internal light modes dynamically and demonstrate a possible way to implement a fully quantum mechanical time-delayed feedback. Our theoretical approach proposes a way to experimentally feedback control quantum correlations in the single photon limit.
In this thesis three different approaches to control the quantum statistics in nanoscopic devices are investigated. In the first part of the thesis, mechanical oscillations in solid state devices are manipulated by an external laser to show bistabilities, coherent amplification, and enhanced damping. The dynamics is controlled by the frequency of the external laser. This study is performed in a semiclassical approximation, where expectation values of operators are factorized. These effects are used to show the similarities and differences between optomechanical systems and phonon cavities. In the regime of coherent amplification of phonons the theoretical description is extended to a fully quantum mechanical model. This reveals additional resonances due to collective processes that can be addressed via the frequency of the external laser. In the second part, a quantum light source is considered for manipulation of the emission statistics of a second quantum optical system serving as target. A Jaynes-Cummings system is used as a source that may be tuned from nonclassical to thermal emission via the pump strength. The target consists of a cavity with two emitters. The nature of the imprinted statistics is studied by higher order correlation functions. The resulting statistics deviates strongly from thermal, coherent or antibunched statistics. The third part considers coherent self feedback with time-delay, which is motivated by classical Pyragas control. The finite time delay is used as control parameter. The matrix product state evolution in the picture of the quantum stochastic Schrödinger equation is employed as a systematic way of dealing with quantum self-feedback. In this thesis, the approach is extended to include higher order terms in the time evolution operator. This is done systematically so that arbitrary order can be included enabling faster numerical evaluation. This extension of the matrix product state evolution method allows to investigate efficiently the emission statistics of a single mode cavity containing two emitters. It is shown that feedback can enhance antibunching in the light field while also counter-intuitively increasing the number of emitted photons at the same time. Expressing the time evolution operator as a sum of sparse matrices allows to include time dependent Hamiltonians in an efficient manner. Considering the pulsed excitation of a two level emitter, it is demonstrated that emission statistics may be tuned via feedback. This allows to switch between single photon emission and enhanced two photon emission.
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.