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Journal of Physics Conference Series

The effective Hamiltonian as a necessary basis of the open quantum optical system theory

Ashat M. Basharov

Published Apr 1, 2021• 10 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/1890/1/012001

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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.
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 We show that a number of particle decay problems, which are described by the non-Hermitian Hamiltonian (correspondingly, the nonunitary dynamics is spoken of), can be correctly and consistently reformulated in terms of stochastic differential equations upon application of the algebraic perturbation theory. In this formulation, kinetic equations are obtained in the standard scheme of the theory of open quantum systems. The parameters of the kinetic equation coincide with similar parameters describing the decay of a particle when the decay is considered on the basis of the non-Hermitian Hamiltonian.
Abstract In the conventional quantum mechanics of conserved systems, Hamiltonian is assumed to be a Hermitian operator. However, when it comes to quantum systems in presence of dissipation and/or noise, including open quantum optical systems, the strict hermiticity requirement is nor longer necessary. In fact, it can be substantially relaxed: the non-Hermitian part of a Hamiltonian is allowed, in order to account for effects of dissipative environment, whereas its Hermitian part would be describing subsystem’s energy. Within the framework of the standard approach to dissipative phenomena based on a master equation for the reduced density operator, we propose a replacement of the hermiticity condition by a more general condition of commutativity between Hermitian and anti-Hermitian parts of a Hamiltonian. As an example, we consider a dissipative two-mode quantum system coupled to a single-mode electromagnetic wave, where we demonstrate that the adjoint-commutativity condition does simplify the parametric space of the model.