Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

The role of discrete orthogonal polynomials in constructing solutions of the dynamical equations describing coherent excitation of quantum systems in the field of laser radiation

Sary Banjak, В. А. Савва

Published Jan 1, 2020• 1 citation
Publisher

Abstract

The paper presents and implements an algorithm for constructing exact analytical solutions of the corresponding differential equations. The solutions are obtained without integration of the equations but with the use of methods of a discrete mathematics. The discrete Fourier transform of the equations and discrete orthogonal polynomials in Fourier space are used. Specially constructed polynomials are used and the solutions of the coherent dynamics equations of some quantum systems excited by laser radiation are given.

DOI
10.1063/5.0000104

Related papers

Coherent Dynamics of Quantum Systems with Non-Uniform Fourier Space Excited by Laser Radiation

Sary Banjak, Vadim Savva

The algorithm is presented to solve dynamical equations for excitation of molecular models with multiple energy levels. It uses only discrete structures: discrete orthogonal polynomials constructed specially in Fourier space of the probability amplitudes, discrete Fourier transform and leads to exact solution of the differential equations and to discrete distribution of the quantum systems by energy levels.

ZANCO Journal of Pure and Applied Sciences•Dec 21, 2022•0 citations

Solving method without integration of some differential equation systems for coherent dynamics of quantum media excited by laser radiation

Sary Banjak

A pragmatic mechanism to obtain exact analytical solutions of equations describing the coherent dynamics of quantum multilevel systems by laser radiation is implemented. The technique shows interesting features and significant connections between different mathematical structures for constructing exact solutions for a large family of quantum systems, and is based on discrete orthogonal polynomials which are used to express the Fourier spectra of the probability amplitudes of an excited quantum system.

Journal of Interdisciplinary Mathematics•Aug 18, 2019•2 citations

On solving coherent dynamics equations with discrete mathematics method for quantum systems under laser excitation

В. А. Савва, Sary Banjak

Molecular coherent excitation calculations are performed using a simple model of quantum N + 1-levels systems. An exact solution of the corresponding system of differential equations is obtained without their integration. For this, the discrete Fourier transform is applied: the sought-for functions – probability amplitudes a[n](t) of a quantum system are represented with Fourier images F[n] (ω) ,i.e. spectra that are described by some corresponding system of discrete orthogonal polynomials. Fourier spectra are calculated using the polynomials constructed. We find the required a[n](t) by calculating the final sum from 0 to N. Based on a one-to-one correspondence: polynomial characteristics vs equations coefficients, we find all the characteristics of quantum systems, the dynamics of which are described by the obtained solution. The construction of various polynomial systems of a discrete variable makes it possible to obtain solutions for quantum systems with various characteristics, including systems with non-equidistant arrangement of energy levels, which are typical for real molecules.

•Jan 1, 2020•0 citations

Orthogonal Polynomials in Analytical Method of Solving Differential Equations Describing Dynamics of Multilevel Systems

В. А. Савва, V. I. Zelenkov, A. S. Mazurenko

An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete analogues. All the obtained solutions are expressed by way of special or elementary functions.

arXiv:math/0101105•Jan 11, 2001•1 citation

Exact analytic solution for coherent dynamics of multilevel Jacobi systems with multiphoton laser excitation

В. А. Савва, V. I. Zelenkov, A. S. Mazurenko

A method based on the use of integral transform and orthogonal polynomials is presented to obtain the analytic solutions of the equations for coherent dynamics of multilevel quantum systems as simple molecular models describing molecule-radiation interactions in the high-power laser field and many other processes. As an example the Jacobi polynomials have been used and exact analytic solution has been obtained for dynamics of the family of multilevel Jacobi systems with both equidistant and nonequidistant energy levels, with various dependencies of dipole transition moments on energy, with both resonant and non-resonant laser excitation. Some special cases of the solution for definite systems from the family are given.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jun 10, 1996•0 citations

Integrable models for quantum media excited by laser radiation: a method, physical interpretation, and examples

В. А. Савва, V. I. Zelenkov

A method to build various integrable models for description of coherent excitation of multilevel media by laser pulses is suggested. Distribution functions over the energy levels of quantum systems depending on the time and frequency detuning are obtained. The distributions follow from Schrödinger equation exact solutions and give the complete dynamical description of laser-excited quantum multilevel systems. Interpretation based on the Fourier spectra of the probability amplitudes of a quantum system is presented. The spectra are expressed in terms of orthonormal polynomials and their weight functions. Matrix elements of the dipole transitions between levels are equal to coefficients of the recurrence formula for the orthonormal polynomial system. Some examples are presented. The Kravchuk oscillator family as an integrable model is constructed to describe the coherent excitation dynamics of multilevel resonance media. It is based on the use of the Kravchuk orthogonal polynomials. The Kravchuk oscillator excitation dynamics is described by the binomial distribution of energy level populations and the distribution parameter depends on excitation conditions. Two known basic models in quantum physics -- the harmonic oscillator and two-level system are the special representatives of the Kravchuk oscillator family.

arXiv:1404.1248•Mar 31, 2014•0 citations
The paper presents and implements an algorithm for constructing exact analytical solutions of the corresponding differential equations. The solutions are obtained without integration of the equations but with the use of methods of a discrete mathematics. The discrete Fourier transform of the equations and discrete orthogonal polynomials in Fourier space are used. Specially constructed polynomials are used and the solutions of the coherent dynamics equations of some quantum systems excited by laser radiation are given.
A pragmatic mechanism to obtain exact analytical solutions of equations describing the coherent dynamics of quantum multilevel systems by laser radiation is implemented. The technique shows interesting features and significant connections between different mathematical structures for constructing exact solutions for a large family of quantum systems, and is based on discrete orthogonal polynomials which are used to express the Fourier spectra of the probability amplitudes of an excited quantum system.
Molecular coherent excitation calculations are performed using a simple model of quantum N + 1-levels systems. An exact solution of the corresponding system of differential equations is obtained without their integration. For this, the discrete Fourier transform is applied: the sought-for functions – probability amplitudes a[n](t) of a quantum system are represented with Fourier images F[n] (ω) ,i.e. spectra that are described by some corresponding system of discrete orthogonal polynomials. Fourier spectra are calculated using the polynomials constructed. We find the required a[n](t) by calculating the final sum from 0 to N. Based on a one-to-one correspondence: polynomial characteristics vs equations coefficients, we find all the characteristics of quantum systems, the dynamics of which are described by the obtained solution. The construction of various polynomial systems of a discrete variable makes it possible to obtain solutions for quantum systems with various characteristics, including systems with non-equidistant arrangement of energy levels, which are typical for real molecules.
A method based on the use of integral transform and orthogonal polynomials is presented to obtain the analytic solutions of the equations for coherent dynamics of multilevel quantum systems as simple molecular models describing molecule-radiation interactions in the high-power laser field and many other processes. As an example the Jacobi polynomials have been used and exact analytic solution has been obtained for dynamics of the family of multilevel Jacobi systems with both equidistant and nonequidistant energy levels, with various dependencies of dipole transition moments on energy, with both resonant and non-resonant laser excitation. Some special cases of the solution for definite systems from the family are given.
A method to build various integrable models for description of coherent excitation of multilevel media by laser pulses is suggested. Distribution functions over the energy levels of quantum systems depending on the time and frequency detuning are obtained. The distributions follow from Schrödinger equation exact solutions and give the complete dynamical description of laser-excited quantum multilevel systems. Interpretation based on the Fourier spectra of the probability amplitudes of a quantum system is presented. The spectra are expressed in terms of orthonormal polynomials and their weight functions. Matrix elements of the dipole transitions between levels are equal to coefficients of the recurrence formula for the orthonormal polynomial system. Some examples are presented. The Kravchuk oscillator family as an integrable model is constructed to describe the coherent excitation dynamics of multilevel resonance media. It is based on the use of the Kravchuk orthogonal polynomials. The Kravchuk oscillator excitation dynamics is described by the binomial distribution of energy level populations and the distribution parameter depends on excitation conditions. Two known basic models in quantum physics -- the harmonic oscillator and two-level system are the special representatives of the Kravchuk oscillator family.