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Moscow University Physics Bulletin

Anharmonic Quantum Oscillator as a Model of Resonant Optics

Ashat M. Basharov

Published Jun 1, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.3103/s0027134925700511

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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.
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.
Abstract Using algebraic perturbation theory, the applicability of the quantum oscillator model within the theory of open quantum systems is demonstrated. The case of resonant and nonresonant interactions between two spatially separated oscillators is considered, where each oscillator interacts with its own thermostat at different temperatures. The possibility of the decay of an isolated oscillator into the thermostat of the oscillator-mediator under conditions of its nonresonant interaction with the latter is demonstrated. In all the cases presented, algebraic perturbation theory does not lead to any nonphysical features or contradictions, such as violations of the second law of thermodynamics or discrepancies in the description of these cases within the frameworks of the global and local approaches.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
Recently, superconducting artificial atom has been justified experimentally as an ideal element for various applications including circuit quantum electrodynamics realizing strong-coupling limit. Some of the proposed systems in this area are usually described in terms of the quantum anharmonic oscillator and display quantum dynamics in macroscopic level, giant nonlinearity as well as very strong coupling with external field and strong coupling with an environment. In this report, we investigate the quantum properties of an artificial atom as a model of multilevel anharmonic oscillator in the framework of photon number distributions as well as the Wigner functions. We concentrate on the regimes of strong driving and giant nonlinearity that allow us to consider artificial atom on the level of few excitation numbers.