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arXiv:1810.00627 · Physica Scripta

The quantum and classical Fano parameter q

Masatomi Iizawa, Satoshi Kosugi, Fumihiro Koike, Yoshiro Azuma

Published Feb 11, 2021• 22 citations
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Abstract

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.

DOI
10.1088/1402-4896/abe580

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•Nov 18, 2015•0 citations
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.
Fano resonance presents an asymmetric line shape formed by an interference of a continuum coupled with a discrete autoionized state. In this paper, we show several simple circuits for Fano resonances from the stable-input impedance mechanism, where the elements consisting of inductors and capacitors are formulated for various resonant modes, and the resistor represents the damping of the oscillators. By tuning the pole-zero of the input impedance, a simple circuit with only three passive components e.g. two inductors and one capacitor, can exhibit asymmetric resonance with arbitrary Q-factors flexiblely. Meanwhile, four passive components can exhibit various resonances including the Lorentz-like and reversely electromagnetically induced transparency (EIT) formations. Our work not only provides an intuitive understanding of Fano resonances, but also pave the way to realize Fano resonaces using simple circuit elements.
The Fano resonance, with a unique asymmetric line shape, is a well-known manifestation of quantum interference with numerous promising applications for light-harvesting devices. Here we report an experimental observation of the Fano effect in an ultracold atom-molecule coupled system realized by photoassociation near a magnetically tunable ddd-wave Feshbach resonance. Our result shows that, compared to ubiquitous single-peak Fano-type line shapes, the coupled Fano resonances feature a two-peak profile. We develop a theory of coupled Fano resonances which agrees very well with the experimental results. This work may open a new perspective for tuning ultracold collisional interaction using the Fano resonance and studying bound states in continuum.
The scattering amplitude from a set of discrete states coupled to a continuum became known as the Fano profile, characteristic for its asymmetric line shape and originally investigated in the context of photoionization. The generality of the model and the proliferation of engineered nanostructures with confined states gives immense success to the Fano line shape, which is invoked whenever an asymmetric line shape is encountered. However, many of these systems do not conform to the initial model worked out by Fano in that (i) they are subject to dissipative processes and (ii) the observables are not entirely analogous to the ones measured in the original photoionization experiments. In this Letter, we work out the full optical response of a Fano model with dissipation. We find that the exact result for the excited population, Raman, Rayleigh, and fluorescence emission is a modified Fano profile where the typical line shape has an additional Lorentzian contribution. Expressions to extract model parameters from a set of relevant observables are given.
The general profile of a resonance, as given by Fano in 1961, is well known throughout physics. This Comment provides some of the simplest derivations of this formula, together with a discussion of subtleties in handling continuum states with regard to their normalization, orthogonality, and completeness. Inadequate appreciation of these subtleties has led to misconceptions and errors in analysis. Recent, novel applications of the Fano formula in condensed matter systems include systems with decoherence and lack of time reversal symmetry which make the resonance profile parameter complex.
We obtain an explicit solution for the stationary state populations of a dissipative Fano model, where a discrete excited state is coupled to a continumm set of states; both excited set of states are reachable by photo-excitation from the ground state. The dissipative dynamic is described by a Liouville equation in Lindblad form and the field intensity can take arbitrary values. We show that the continuum states population as a function of laser frequency can always be expressed as a Fano profile plus a Lorentzian function with effective parameters whose explicit expressions are given in the case of a closed system coupled to a bath as well as for the original Fano scattering framework. Although the solution is intricate, it can be elegantly expressed as a linear transformation of the kernel of a 4 × 4 matrix which has the meaning of an effective Liouvillian. We unveil key notable processes related to the optical nonlinearity and which had not been reported to date: electromagnetic induced-transparency, population inversions, power narrowing and broadening, as well as an effective reduction of the Fano asymmetry parameter. The original Fano model was introduced by U. Fano in 1961 [1] to explain the asymmetric profile that was obtained when monitoring the photo-current as a function of the laser frequency used to ionize a gaz of Helium-like atoms. In parallel, Friederichs [2] developed the mathematical formalism to describe the essential feature of the Fano model: a discrete state coupled to a continuum set of states; both sets of states being reachable by photo-excitation from the ground state. The resulting photo-current, which is proportional to the population of the continuum set of states, as a function of the excitation laser frequency ω L is known as the Beutler-Fano or Fano profile: f (q) = (q + 2 /(2 + 1), where q is the ratio of the transition dipole moment of the ground-discrete and ground-continuum transitions, and = (ω L − ω e)/γ where e is the energy of the discrete state relative to the ground state, ω L is te radiation field frequency, and = nπV 2 is the linewidth of the excited state, induced by its coupling (per unit of energy) nV 2 to the continuum set of states, n being the density of states. Two extensions of the original model have been considered: inclusion of incoherent relaxation and dephasing processes and high field intensities. The motivation to include incoherent processes was first to describe the pressure broadening [3] due to elastic collisions, laser phase fluctuations [4] and spontaneous emission [5, 6]. Nowadays , Fano profiles in nanoscale structures are standard [7, 8], for example in plasmonic nanostructures [9–11], quantum dots, decorated nanoparticles [12] and spin filters [13]. The coherent coupling with the incident light induces large Rabi frequencies which in turn compete with the relaxation rates in order to modify the stationary state. The ability to predict the lineshape, and in general to investigate non linear optical phenomena in