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Moscow University Physics Bulletin

Algebraic Resonance Perturbation Theory in Problems of Nonlinear and Quantum Optics

A. M. Basharov

Published Oct 1, 2025• 2 citations
Publisher

Abstract

Abstract The algebraic resonance perturbation theory (ARPT) is formulated, general expressions of the theory are written down, by means of which effective Hamiltonians are concisely derived, interference interactions are described, and kinetic equations are obtained for various cases of interaction between electromagnetic radiation and quantum systems. The natural integration of the method of stochastic differential equations into ARPT and the physical justification of the principles for the selection of ARPT terms are indicated in the case of approximating the environment of an open system by delta-correlated noise. The inevitability of the emergence of interference interactions is emphasized due to the formulated ARPT requirements regarding the selection of only slowly time-varying terms in the Dirac picture into the effective Hamiltonian.

DOI
10.3103/s0027134925701012

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Abstract The algebraic resonance perturbation theory (ARPT) is formulated, general expressions of the theory are written down, by means of which effective Hamiltonians are concisely derived, interference interactions are described, and kinetic equations are obtained for various cases of interaction between electromagnetic radiation and quantum systems. The natural integration of the method of stochastic differential equations into ARPT and the physical justification of the principles for the selection of ARPT terms are indicated in the case of approximating the environment of an open system by delta-correlated noise. The inevitability of the emergence of interference interactions is emphasized due to the formulated ARPT requirements regarding the selection of only slowly time-varying terms in the Dirac picture into the effective Hamiltonian.
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.
Annalen der Physik , Ser. 5, 9: 338–46 (1931). Received 25 February 1931. The Dirac theory of radiation gives a satisfactory account of the absorption, emission, and dispersion of radiation by atoms. Beyond this, quantum electrodynamics also provides a satisfactory treatment of interference phenomena. The results of the theory in most cases agree with what is expected from the correspondence principle, but the calculations which lead to this objective have hitherto been rather complicated, and the simplest consequences of classical radiation theory can only be derived along the circuitous route through a quite nontransparent Schrödinger equation in an infinite-dimensional space. Below we shall describe a method for treating radiation problems which connects, much more closely than the previous ones, the intuitive conceptions of classical theory with those of wave mechanics and which therefore, in most cases, and without detours, yields the result expected in terms of the correspondence principle. In this method, the starting point shall not be the Hamiltonian function or the corresponding Schrödinger equation in configuration space but rather the equations of motion, i.e. the Maxwell and Dirac wave equations. These differential equations are integrated explicitly, after the value of the wave functions ( E, H , and ψ σ ) at time t = 0 are regarded as given; here it should be noted that the wave functions at time t = 0 are noncommuting quantities. This noncommutability of the initial values of E, H , and ψ σ does not disturb the integration, however, as long as the interaction between matter and radiation is assumed to be small and only linear terms in this interaction are considered.
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.
A molecule in the presence of a radiation field is described by a non-Hermitian reduced effective Hamiltonian and its eigenvectors, the effective system states. These states form a natural basis for the evaluation of spectroscopic properties; its use requires a generalized formulation of quantum dynamics. The mathematical structure of the vector space defined by those states is investigated; that vector space is found to be a Banach space which reduces to Hilbert space if the energies of the effective system states become real. Time reversal symmetry is found to be an important feature of the overall problem. The generalized time development operator and the Schrödinger equation which defines it are deduced, and the effect of that operator on the effective system states is discussed. The proper generalization of time-dependent perturbation theory in the interaction picture is then derived and its use in applications is discussed in general terms.
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.