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Physics-Uspekhi

On the question of a classical analog of the Fano problem

Mikhail V. Lebedev, O. V. Misochko

Published Nov 1, 2021• 1 citation
Publisher

Abstract

A system of interacting classical oscillators is discussed, which is an analog of the quantum-mechanical problem of a discrete energy level interacting with an energy quasicontinuum of states considered by Fano.The limiting transition to the continuous spectrum and the possible connection of this problem with the generation of coherent phonons are analyzed.

DOI
10.3367/ufne.2021.11.039098

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A system of interacting classical oscillators is discussed, which is an analog of the quantum-mechanical problem of a discrete energy level interacting with an energy quasicontinuum of states considered by Fano.The limiting transition to the continuous spectrum and the possible connection of this problem with the generation of coherent phonons are analyzed.
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We seek to illustrate Fano interference in a classical coupled oscillator by using classical analogues of the atom–laser interaction. We present an analogy between the dressed state picture of coherent atom–laser interaction and a classical coupled oscillator. The Autler–Townes splitting due to the atom–laser interaction is analogous to the splitting of normal-mode frequencies of a coupled oscillator. Using this analogy, we simulate and experimentally demonstrate Fano interference and the associated phenomena in three-level atoms in a coupled electrical resonator circuit. This work aims to highlight analogies between classical and quantum systems for students at the postgraduate and graduate levels. Also, the reported technique can be easily realized in undergraduate laboratories.
In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned.
Abstract The Fano resonance has been a familiar and important feature in atomic and molecular physics for more than half a century. Typically, the combination of a discrete state with one or more continua results in an asymmetric peak in the ionization spectrum. The peak-shape, called the Fano profile, can be expressed by a simple formula derived by Fano in 1935. However, the interpretation of its characteristic parameter q , which represents the asymmetry of the peak in the formula, is not necessarily intuitively transparent. The Fano resonance is not necessarily a quantum effect, but it is a manifestation of a certain physical mechanism in various systems, both quantum and classical. Through the derivation of q from the known classical pictures with the classical coupled oscillator, we interpret the q value with a geometrical view. And further, we introduce a complex valued q parameter for the description of the resonance with a damped oscillator.
In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned. If a diffusive environment is considered, we conclude that the "lifetime" of discreteness is approximately 1/κ1/κ1/κ (κκκ is the diffusion constant), thus it was possible to relate the classical limit of a spectra with the action of an environment and experimental resolution.