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Journal of Experimental and Theoretical Physics

“Global” and “Local” Approaches to the Theory of Open Quantum Optical Systems

Ashat M. Basharov

Published Nov 1, 2020• 27 citations
Publisher

Abstract

The application of “exact/complete” initial Hamiltonian of an open optical system and its surroundings and “approximate” effective Hamiltonian for the derivation of the kinetic equation of the open optical system is analyzed in the conditions of the Markovian approximation and the representation of the open system surroundings (thermostat) as a delta-correlated noise. It is shown that the characteristic time hierarchy of the open system naturally necessitates a transition from the aforementioned exact Hamiltonian to the approximate effective Hamiltonian for the subsequent use of the Markovian approximation and the model of delta-correlated surroundings of the open system in the local approach. The Schrödinger equation for the wavevector of the open system and its surroundings is a quantum stochastic differential equation, from which the kinetic equation describing both familiar and new results can be derived easily in the standard manner.

DOI
10.1134/s1063776120110011

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The application of “exact/complete” initial Hamiltonian of an open optical system and its surroundings and “approximate” effective Hamiltonian for the derivation of the kinetic equation of the open optical system is analyzed in the conditions of the Markovian approximation and the representation of the open system surroundings (thermostat) as a delta-correlated noise. It is shown that the characteristic time hierarchy of the open system naturally necessitates a transition from the aforementioned exact Hamiltonian to the approximate effective Hamiltonian for the subsequent use of the Markovian approximation and the model of delta-correlated surroundings of the open system in the local approach. The Schrödinger equation for the wavevector of the open system and its surroundings is a quantum stochastic differential equation, from which the kinetic equation describing both familiar and new results can be derived easily in the standard manner.
The hierarchy of characteristic times naturally justifies the necessity of the transition from the initial “exact” Hamiltonian of an open quantum system and its environment to an approximate effective Hamiltonian for the further use of the Markov approximation and the model of the delta-correlated environment of the open system. The transition to the effective Hamiltonian within the algebraic perturbation theory makes it possible to take into account interference relaxation channels and specific interference of random processes that cannot be detected and are absent in the rotating wave approximation. It has been shown that specific interference in the case of an ensemble of identical quantum oscillators results in a non-Wiener dynamics of the ensemble.
Abstract Based on a model open system, the kinetic equation obtained from the initial exact Hamiltonian with anti-rotating terms is shown to differ from the kinetic equation from the approximate effective Hamiltonian without anti-rotating terms. The paper proves necessity of transition from the exact Hamiltonian to the effective Hamiltonian, with the main idea of its construction being the requirement for the absence of rapidly varying terms in the interaction representation. This requirement is necessary to obtain the correct kinetic equation.
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 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.
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.