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Journal of Physics A Mathematical and General

Allowed transitions between non-adjacent levels and analytical solutions for multilevel quantum systems

Alexander S Mazurenko

Published Oct 31, 2001• 0 citations
Publisher

Abstract

An analytical method is formulated to solve the Schrödinger equation for a number of multilevel systems with dipole transitions between adjacent and non-adjacent levels in multifrequency fields. It is shown that orthogonal polynomials can be used to construct exact analytical solutions. The excitation dynamics of the three non-equidistant levels system is analysed in detail.

DOI
10.1088/0305-4470/34/44/312

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An analytical method is formulated to solve the Schrödinger equation for a number of multilevel systems with dipole transitions between adjacent and non-adjacent levels in multifrequency fields. It is shown that orthogonal polynomials can be used to construct exact analytical solutions. The excitation dynamics of the three non-equidistant levels system is analysed in detail.
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An excitation dynamics of new quantum systems of N equidistant energy levels in a monochromatic field has been investigated. To obtain exact analytical solutions of dynamic equations an analytical method based on orthogonal functions of a real argument has been proposed. Using the orthogonal Legendre functions we have found an exact analytical expression for a population probability amplitude of the level n. Various initial conditions for the excitation of N-level quantum systems have been considered.
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.
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.