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Bulletin of the Russian Academy of Sciences Physics

Nonlinear Effects in an Ensemble of Anharmonic Oscillators

Ashat M. Basharov

Published Dec 1, 2024• 1 citation
Publisher

Abstract

Abstract The author considers three types of resonance of a classical coherent electromagnetic wave with the same carrier frequency as a quantum anharmonic oscillator, based on algebraic perturbation theory. The corresponding effective operators of the dipole moment of the anharmonic oscillator are obtained, allowing the polarization of the excited medium under the considered resonance conditions to be calculated. Some possibilities are established for the creation of nonlinear effects.

DOI
10.1134/s1062873824708407

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Abstract The author considers three types of resonance of a classical coherent electromagnetic wave with the same carrier frequency as a quantum anharmonic oscillator, based on algebraic perturbation theory. The corresponding effective operators of the dipole moment of the anharmonic oscillator are obtained, allowing the polarization of the excited medium under the considered resonance conditions to be calculated. Some possibilities are established for the creation of nonlinear effects.
Abstract The specific features of the theory of resonant interactions of electromagnetic coherent fields with an anharmonic oscillator, as compared to the case of a multilevel atom, are highlighted. It is shown that within the framework of the local approach—the algebraic resonance perturbation theory—it is possible to account for the requirements of the global approach of the theory of open quantum systems. Based on the auxiliary model of the ‘‘internal nonlinearity’’ of the anharmonic oscillator, the possible conditions for the resonance of a coherent wave with an anharmonic oscillator involving the absorption of a single photon are analyzed.
The speci c features of the theory of resonant interactions of electromagnetic coherent elds with an anharmonic oscillator, as compared to the case of a multilevel atom, are highlighted. It is shown that within the framework of the local approach - the algebraic resonance perturbation theory - it is possible to account for the requirements of the global approach of the theory of open quantum systems. Based on the auxiliary model of the «internal nonlinearity» of the anharmonic oscillator, the possible conditions for the resonance of a coherent wave with an anharmonic oscillator involving the absorption of a single photon are analyzed.
The propagation effects of linearly polarized near-resonant laser radiation through a thermal ensemble of Doppler-broadened anharmonic oscillators are considered. The dynamical equations of motion are analytically solved by a perturbation approach to obtain the nonlinear polarization. Subsequent self-consistent analysis of the Maxwell wave equation then leads to a classical area theorem, which is in qualitative agreement up to third order with the corresponding quantum mechanical result. There is, however, a discrepancy between the respective coefficients of the third-order terms.
Resonant interactions of a coherent field with an anharmonic oscillator are classified based on the established analogy with multiphoton interactions of light with atoms. It is shown that interference interactions in which the absorption of a single coherent field quantum induces a multiphoton transition in the anharmonic oscillator are superpositions of resonances of the “cascade” and “lambda-V” types, and a general description of these resonances is given. For the third- and fourth-order nonlinearities, the general expressions are applicable to a resonant process with the absorption of a single photon and the double excitation of the oscillator. Resonant processes of the second harmonic generation and “rectification” of radiation are predicted and described; experimental investigation of these processes will provide information about anharmonicity parameters in the proposed model.
We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.