2007, Vol.14, pp.152-159

In the method with the use of integral transform with orthogonal polynomials to construct exact analytical solutions for dynamics of quantum multilevel systems in laser field algorithm is presented to solve dynamical equations describing excitation by laser pulse with an arbitrary prescribed form. Examples of solutions are given.
It is justified that orthogonal polynomials are adequate and natural instruments for analytical investigation of the dynamics of multilevel quantum systems since orthogonal polynomials and probabilities amplitudes of dynamical equations are connected to one another with Fourier transform.
A brief survey of the theory of q-calculus, the theory of special q-functions and orthogonal q-polynomials as special cases of basic hypergeometric functions is given. Certain orthogonal q-polynomials being q-deformed analogues of classical orthogonal polynomials are presented. Orthogonal q-polynomials are promising mathematical structures for constructing new multilevel quantum systems and for obtaining exact analytical solutions describing their coherent dynamics in laser fields and for other physical problems as well.
Key words: ultrashort laser pulses, coherent laser excitation, multilevel quantum systems, exact analytical solutions, orthogonal q-polynomials

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