An original general approach to the description of open quantum systems in the Markovian approximation was formulated for the example of the dynamics of high-Q cavity mode photons in the presence of nonresonant cavity atoms bound with an external vacuum electromagnetic field.
It is shown that the effective Hamiltonian representation, as it is formulated in author’s papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are “locked” inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
Abstract This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.
For a model of an open quantum system—a concentrated ensemble consisting of similar atoms and interacting with a one-dimensional quantum vacuum environment with a zero photon density—quantum stochastic differential equations of a non-Wiener type of the general form have been obtained; based on the equations, kinetic equations describing a wide class of physical systems are derived. The distinctive feature of such systems is effects of suppression of collective spontaneous emission and stabilization of the excited state. For the open classical system exposed to the action of noise in the form of a Levy process of the general non-Gaussian kind, kinetic equations of the Fokker-Planck type with fractional derivatives have been obtained based on classical non-Wiener stochastic differential equations. This emphasizes the common base of the developed theory for different types of open systems, which is expressed in using the mathematical formalism of stochastic differential equations of the general non-Wiener type.
Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges have been introduced, originating from different fields, such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations. This diversity, while indicative of the field's relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? Here we give a systematic overview and concise discussion addressing these questions. We make use of a unified framework which very conveniently allows to link different schemes and, this way, may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space which in a minimal fashion includes effective reservoir modes. This in turn offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.
Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.
The interaction of a quantum system with a many degree of freedom environment entails damping and the loss of quantum coherence. In these two lectures we focus on two aspects of open quantum system dynamics. First, we present decoherence as a dynamical phenomenon and determine the corresponding time scales, discuss an experiment and the nature of robust states, and mention universal properties of short-time decoherence of macroscopic superpositions. Secondly, in analogy to the classical description of open systems in terms of Langevin equations, we present a stochastic approach to open quantum system dynamics in terms of a stochastic Schrödinger equation. We give a microscopic derivation that allows us to generalize the stochastic approach to non-Markovian situations. We discuss a convolutionless formulation of our theory and finally, we focus on the quantised version of Brownian motion as an often-employed example.