The quantum and classical Fano parameter q
Masatomi Iizawa, Satoshi Kosugi, Fumihiro Koike, Yoshiro Azuma
TThe quantum and classical Fano parameter q Masatomi Iizawa , Satoshi Kosugi , Fumihiro Koike andYoshiro Azuma Department of Materials and Life Sciences, Sophia University, Tokyo 102-8554,JapanE-mail: [email protected]
Abstract.
The Fano resonance has been regarded as an important phenomenon inatomic and molecular physics for more than half a century. Typically, a combinationof one quantum bound state and one or more continuum result in an asymmetric peakin the ionization spectrum. The peak-shape, called the Fano profile, can be expressedby the simple formula derived by Fano in 1935. However the interpretation of itsmain parameter q , which represents the asymmetry of the peak in the formula, is notintuitively transparent. The Fano resonance is not necessarily a quantum effect, but itis a manifestation of a certain physical mechanism in various systems, both quantumand classical. We present three intuitively transparent classical pictures and rigorouslyderive their Fano profiles to properly formulate the physics of the Fano parameter. Keywords : Fano resonance, double photoexcitation, classical mechanics, scienceeducation, history of science a r X i v : . [ phy s i c s . a t o m - ph ] O c t he quantum and classical Fano parameter q Figure 1.
Helium photoabsorption spectroscopy around a two-electron excitationstate 2 s p P ◦ (cited from Madden (1964) [15])
1. Introduction
Typical excitation resonances in atomic physics have the well-known symmetric peak-shapes called the Lorentzian, Breit-Wigner, and Cauchy distribution. However, thehelium double photoexcitation to 2 s p P ◦ , the first experiment performed utilizingsynchrotron radiation in the late sixties, presented a dramatic demonstration ofasymmetric peaks in the photoabsorption spectrum. (Figure 1). (A historical notecan be found in Appendix A and recent examples in Appendix B.) The first theoreticalformulation of this ”Fano resonance” was developed some time before by Fano (1935) [1]and then extended and refined by Fano (1961) [2] after the photoexcitation experiment.In the latter paper, it was argued rigorously that from the superposition state consistingof a discrete state ϕ and a continuum ψ E (cid:48) Ψ E = aϕ + (cid:90) dE (cid:48) b E (cid:48) ψ E (cid:48) ( a (cid:54) = 0) (1)(Fano (1961) [2] equation (2)), the Fano profile formula(total scattering cross-section) ∝ ( q + (cid:15) ) (cid:15) = 1 + q − q(cid:15) (cid:15) (2)could be derived. Here, (cid:15) is the photon energy offset from the peak position andnormalized by a half of the resonance width, called the reduced energy (Fano (1961) [2]equation (21)). Figure 2 plots this formula. The curve has a clear asymmetricpeak with the minimum value going down to zero. (Note: If the state includestwo or more continuum states, this spectrum does not necessarily go down to zero(Fano (1961) § he quantum and classical Fano parameter q C r o ss - s e c t i on σ ( ε ) Reduced Energy ε q=0q=1q=2 C r o ss - s e c t i on σ ( ε ) Reduced Energy ε q=5q=10q=20 Figure 2.
Fano profile, graphical plots of Equation (2) for several values of q . theory and practical utility. However, some of the important concepts in Fano’s paper [2]are not necessarily obvious. These include the basic meaning of the q parameter (alsocalled the profile index or Fano parameter) or an explanation of why the peak becomesasymmetric when a bound state is superposed on continuum. To illustrate the physicalessence of the Fano resonance, we first discuss it in a classical system instead of aquantum system. Fano resonances are often regarded as a phenomenon specific toquantum mechanics, but they also occur in classical mechanics.Recent analytical studies by Riffe (2011) [5] argued that Fano resonances occur incertain classical linear systems. However, Riffe’s paper overlooks the fact that the Fanoresonance is an ubiquitous phenomenon among various physical systems rather thana concept valid only for a particular system. Some authors introduced the so-called“classical Fano resonance” as distinct from the (quantum) Fano resonance [4] [6] [7] [8].Also, a similar line shape in the classical coupled oscillator was noted [3]. They basedthe analogy in terms of the similar peak-shapes. We argue for the nature of these Fanoresonance in terms of the physical mechanism instead of just looking for analogy ormetaphor in terms of the peak-shape. To demonstrate this, we analytically derive theFano profile from two classical models. The first “coupled oscillator” model itself wasdiscussed by [4] [6] [7] [8]. Then, we will introduce a second, original physical model thatwe formulated based on a one spring model. The latter is advantageous for graspingonly the most fundamental essence of the Fano resonance without any additional details.
2. Classical Fano resonance in the coupled oscillator system
We have normally only a single resonance frequency in a damping simple harmonicoscillator. To discuss the interplay of the two resonance oscillations that residesclose each other in resonance frequencies, we consider the system consisting of weaklyconnected two damping simple oscillators. We illustrate the system in figure Figure 5;Two oscillators of mass m and m with the stiffness constant k and k are connected he quantum and classical Fano parameter q (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Figure 3.
Coupled harmonic oscillators with two springs and two masses by a weak spring with stiffness K . If the values of k and k are close and their dampingconstants are enough large, the oscillations of two harmonic oscillators will overlap andprovide us with a conspicuous anomaly in the resonance features. In the present paper,we try to consider an extreme case in which the damping factor of m , γ is zero; thecase γ (cid:54) = 0, and γ = 0.The sharp one occurs on the shoulder of the broad one. We call the sharp resonancesdiscrete states, since the amplitude is large only in the narrow vicinity of the peak andvery small elsewhere. This sharp resonance is superimposed on the shoulder of a broadresonance whose amplitude A varies gently as the external frequency ω is changed.At least two springs and two masses are required to produce two resonances. Let usconsider the system depicted in Figure 3. Hereafter, we set the masses m = m = 1, thedisplacement x , x , the damping coefficients γ (cid:54) = 0, γ = 0 and the spring constants K , k .To solve the equation of motion of this system,¨ x + γ ˙ x + K ( x − x ) = F cos ωt (3)¨ x + γ ˙ x + k x − K ( x − x ) = 0 (4)is expressed as x ( t ) = c ( ω ) cos ωt, (5)and the amplitude c of x is c ( ω ) = ω − ω + iγ ω ( ω − ω + iγ ω )( ω − ω + iγ ω ) − v F (6)where ω = √ K, ω = (cid:112) k + K, v = − K. (7)We use the variable v for consistency with later discussion. Plotting | c ( ω ) | , which isthe absolute square of amplitude c , resulting in Figure 4.The square of the frequency ω is placed on the horizontal axis because the energy E of a classical harmonic oscillator (one spring and one mass) is represented by E = 12 mω . (8)Thus, the horizontal axis is the energy provided by an external force. Incidentally, theFano profile can be derived as a vibration frequency as well as energy on a horizontalaxis. In a quantum system, it is clear that it can be discussed either way because E = (cid:126) ω , but even in a classical system, the Fano profile can be derived with eitherenergy ∝ ω or frequency ω as the horizontal axis. he quantum and classical Fano parameter q | c ( ω ) | ω Figure 4.
A plot of the square | c ( ω ) | of the norm of the amplitude of x of the systemof the two springs and the two masses (Figure 3) respect to ω , which is the square ofthe frequency of the external force (where ω = 0 . , ω = 0 . , v = 0 . , γ = 0 . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Figure 5.
Damped coupled oscillator
The vertical axis represents the square of the norm of amplitude | c ( ω ) | becausethe energy of the classical harmonic oscillator E (one spring and one mass) becomes E = 12 kA . (9)In other words, the vertical axis will depend on the energy of the mass with displacement x . Taking these factors into consideration, Figure 4 shows the energy given by theexternal force transmitted to mass 1.In Figure 4, two resonances can be seen. Both peaks are broad here. We would likefind the condition that makes one of them sharper, coming closer and overlapping theshoulder of the other peak that remains broad.In order to realize this, we can consider the coupled oscillator which uses threesprings and two masses (Figure 5), introduced by [4] [6] [7]. The equations of motionare similar to the previous one, with only one spring added,¨ x + γ ˙ x + ω x + v x = F cos ωt (10)¨ x + γ ˙ x + ω x + v x = 0 (11)where ω = (cid:112) k + K, ω = (cid:112) k + K, v = − K. (12)Compared to the previous two springs model, the frequency changes to ω = √ k + K .The new spring with spring constant k , can reduce the amplitude, bring two peaks closerwhile keeping one broad shoulder, the other one sharp and overlapping the shoulder.Substituting the solution x ( t ) = c ( ω ) cos ωt, (13) he quantum and classical Fano parameter q c becomes c ( ω ) = ω − ω + iγ ω ( ω − ω + iγ ω )( ω − ω + iγ ω ) − v F . (14)Now, we plot | c ( ω ) | against ω (Figure 6 (left side)). As before, this graph showshow the energy provided by the external force is transmitted to mass 1. Looking atFigure 6 (left side), two peaks are found, ω the lower broader (called peak a , withangular frequency ω a ) spread with a Lorentian shape, whereas the higher narrower peak ω (Called peak b , with angular frequency set as ω b ) is very sharp and can be found atthe foot of peak a .Expansion of peak b looks like Figure 6 (right side). The shoulder of the gentlepeak suddenly drops to 0 near ω = ω = 1 .
44, and after the sharp peak, it turnsback into a gentle slope. It can be said that this was caused by the gentle shoulder(continuum state) interfering with the sharp peak (discrete state). The asymmetricresonance phenomenon caused by such interaction between the discrete state and thecontinuum state is called Fano resonance. q In the last section, we reviewed and provided interpretion for previous discus-sions [4] [6] [7] [8] . Now, we increase the attenuation coefficient γ without changingany other parameter. Figure 6 has the same attenuation coefficient γ = 0 .
025 as before,and those with γ = 1 .
41 and γ = 10 in Figure 7 and 8.Comparing the figure on the right of Figure 6–8 with the Fano profile (Figure 2),it is clearly seen that the attenuation coefficient γ is related to the Fano coefficient q in Fano profile’s formula σ ( (cid:15) ) = ( q + (cid:15) ) (cid:15) = 1 + q − q(cid:15)a + (cid:15) . (15)In the next section the Fano profile will actually be derived from this classical system. For the coupled oscillator (Figure 3) represented by the formula Equation (10), Equation(11), we calculate | c ( ω ) | = c ( ω ) c ∗ ( ω ) (16)in the vicinity of peak b using the displacement amplitude of mass 1 c expressed byEquation (14). For simplicity, we set γ = 0 as in the previous discussions.When diagonalizing the equations of motion Equation (10) and Equation (11) ofthe previous system, It can be seen that the resonance frequencies are somewhat shiftedfrom the natural frequencies ω , ω . The case at Figure 6, shows it visually. We call thesepeaks peak a and peak b respectively as in the previous discussion. The diagonalized he quantum and classical Fano parameter q | c ( ω ) | ω | c ( ω ) | ω Figure 6.
Coupled oscillator (where ω = 1 . , ω = 1 . , γ = 0 . , γ = 0 , v = 0 . γ = 0 . | c ( ω ) | ω | c ( ω ) | ω Figure 7. γ = 1 . | c ( ω ) | ω | c ( ω ) | ω Figure 8. γ = 10 result is, ω a = ω − v ω − ω (17) ω b = ω + v ω − ω , (18)which shows that both peaks are shifted by the same amount in opposite directions. he quantum and classical Fano parameter q b position is adjusted by varying Equation (14) so that˜ (cid:15) def = ω − ω b = ω − ω − v ω − ω , (19)comes down to zero. Now, we used the symbol ˜ (cid:15) because this variable indicates energy,recalling that E = 12 mω (20)is the energy of the classical harmonic oscillator.In the expression Equation (19) since ˜ (cid:15) = ω − ω b = ( ω + ω b )( ω − ω b ), we canregard ( ω + ω b ) as a constant if we consider it in the narrow neighborhood of ˜ (cid:15) = 0.Therefore, since it ˜ (cid:15) ∼ ( ω − ω b ), the same argument holds for the energy ((the constantmultiplication of) the square of the frequency) even though the first power of thefrequency is considered.Next follows the expression for the amplitude c ( ω ) (Equation (14)) with ˜ (cid:15) (Equation (19)). We replace the numerator and denominator of c ( ω ) except for F by A, B (in brief, c ( ω ) = AB F ). The numerator A = ω − ω becomes A = ω − ω (21)= − ˜ (cid:15) − v ω − ω . (22)Further transformation would yield A = − γ ω v ( ω − ω ) (cid:20) γ ω ( ω − ω ) v ˜ (cid:15) + 1 γ ω ( ω − ω ) (cid:21) . (23)The denominator B = ( ω − ω + iγ ω )( ω − ω + iγ ω ) − v can be separated into thereal part and the imaginary part using ˜ (cid:15) , then (cid:60) ( B ) = ( ω − ω )˜ (cid:15) (24) (cid:61) ( B ) = − γ ω (cid:18) ˜ (cid:15) + v ω − ω (cid:19) . (25)Now we are examining the properties in the neighborhood of ˜ (cid:15) = 0 . v ω − ω (cid:29) ˜ (cid:15) ≈ (cid:61) ( B ) ≈ − γ ω v ω − ω . (27)This is the only approximation employed in this derivation. In the neighborhood thepeak b , the real part and the imaginary part can be combined as, B ≈ γ ω v ω − ω (cid:20) γ ω ( ω − ω ) v ˜ (cid:15) − i (cid:21) . (28)Summarizing the above, the amplitude c ( ω ) can be written as c ( ω ) ≈ − γ ω v ( ω − ω ) (cid:104) γ ω ( ω − ω ) v ˜ (cid:15) + γ ω ( ω − ω ) (cid:105) γ ω v ω − ω (cid:104) γ ω ( ω − ω ) v ˜ (cid:15) − i (cid:105) F . (29) he quantum and classical Fano parameter q γ ω ( ω − ω ) v ˜ (cid:15) ., proportional to the energy. (cid:15) def = 1 γ ω ( ω − ω ) v ˜ (cid:15) (30)Next, define q as q def = 1 γ ω ( ω − ω ) . (31)Then, the amplitude Equation (29) can be written as c ( ω ) ≈ − (cid:15) + q(cid:15) − i F ω − ω . (32)Therefore, | c ( ω ) | can be written as | c ( ω ) | ≈ ( (cid:15) + q ) (cid:15) + 1 F ( ω − ω ) . (33)Since F ( ω − ω ) is a constant, | c ( ω ) | is a constant multiple of ( (cid:15) + q ) (cid:15) +1 . | c ( ω ) | ∝ ( (cid:15) + q ) (cid:15) + 1 (34)This is nothing but the Fano Profile itself. That means q is the Fano parameter, whichproduces the Figure 2.From the above discussion, it is clear that the attenuation coefficient γ is inverselyproportional to the Fano parameter q . q ∝ /γ (35)Damping reduces the energy of the system. Therefore, it can be said that Fanoparameter q is an indicator of how much energy the system is retained. When q isinfinite, the energy of the system does not leak. Under this situation, the Fano profileturns into the Lorentzian. On the other hand, when q is close to 0, the energy of thesystem dissipates rapidly. In this case, the Fano profile becomes a dip looking likean upside-down Lotentzian. This is called the Window Resonance [9] in absorptionspectroscopy.
3. Classical Fano resonance in a system employing only one spring
The Fano resonance caused by the superposition of continuum states and a discretestate becomes a Lorentzian as q → ∞ . In other words, the superposition of continuumstates reduces q from infinity. We can give an interpretation to the Fano parameter q illustrating the gradual deformation of the lorentzian lineshape to Fano line-shape fora harmonic oscillator system (one spring and one mass) with mechanical perturbationintroduced. he quantum and classical Fano parameter q σ ε Figure 9.
The amplitude of a driven damped one spring and one mass oscillator (inthe neighborhood of the peak)
We start the discussion with the simple case of a damped driven oscillator whichexhimbits a near Lorentzian profile. Its equation of motion can be written as¨ x + 2 γ ˙ x + ω x = F cos ωt (36)where γ is the attenuation factor and F cos ωt is the external force.Let us consider the following simple Lorentzian profile (probability density functionof standard Cauchy distribution) (Figure 9) σ ( (cid:15) ) = 1 (cid:15) + 1 . (37) The phase difference between the external force and the displacement of the mass(abbreviated as “the phase difference”) is given by θ ( ω ) = tan − (cid:18) γωω − ω (cid:19) (38)in a classical harmonic oscillator. The phase difference and the angular frequencycorrespond one to one, and the phase difference can be defined in the interval − π/ π/ ω tothe phase difference θ between the external force and mass displacement.Considering this simplest Lorentzian, the correspondence between the phasedifference and the angular frequency takes a simple form θ ( ω ) = tan − (cid:18) − ω (cid:19) (39)hence tan θ ( ω ) = − ω (40) he quantum and classical Fano parameter q σ δ ε Figure 10.
The amplitude of a driven damped one spring and one mass oscillator interms of the phase difference then cot θ ( ω ) = − ω. (41)Let’s represent the Lorentzian with phase difference by defining the new variable δ (cid:15) (0 < δ (cid:15) < π ) satisfying (cid:15) = − cot δ (cid:15) . σ ( (cid:15) ) = 1 (cid:15) + 1 (42)= 1cot δ (cid:15) + 1 (43)= sin δ (cid:15) cos δ (cid:15) + sin δ (cid:15) (44)= sin δ (cid:15) (45)With phase difference the Lorentzian is represented by a simple expression(Figure 10).What happens if this phase difference δ (cid:15) is perturbed? Let’s subtract the constant δ q , from the phase difference and include it in the expression for the Lorentzian as shownin Equation (45).sin ( δ (cid:15) − δ q ) = (sin δ (cid:15) cos δ q − cos δ (cid:15) sin δ q ) (46)= (sin δ (cid:15) cos δ q − cos δ (cid:15) sin δ q ) × δ (cid:15) cos δ q − cos δ (cid:15) sin δ q ) (sin δ (cid:15) + cos δ (cid:15) )(sin δ q + cos δ q ) (48)= (cot δ (cid:15) − cot δ q ) (1 + cot δ (cid:15) )(1 + cot δ q ) (49)= ( (cid:15) − cot δ q ) (1 + (cid:15) )(1 + cot δ q ) (50)From the expression Equation (41) showing the correspondence between the phasedifference and the angular frequency of one spring and one mass system, it is he quantum and classical Fano parameter q δ =0 δ = π /2 δ = π δ =3 π /2 * viewpointsin ( δ ) Figure 11. sin ( δ ) on cylindrical coordinate understood that the part cot δ q depends on the angular frequency. Therefore, withthe transformation by q def = − cot δ q , we can return to angular frequency picture again.sin ( δ (cid:15) + δ q ) = ( (cid:15) + q ) (1 + (cid:15) )(1 + q ) (51)= 11 + q ( q + (cid:15) ) (cid:15) (52)Since q is a constant, q is also a constant. We find that this expression is the Fanoprofile itself. (Figure 2)sin ( δ (cid:15) + δ q ) ∝ ( q + (cid:15) ) (cid:15) (53) We proceed with the development of the geometrical interpretation from the previoussection. We write the square of sine curve sin ( δ ) on cylindrical coordinate (Figure 11).Viewing from the origin towards the direction of the δ = − π/ (cid:15) = − cot δ = tan( δ − π/
2) (Figure 12).Next, by shifting the viewing angle by δ q , the Fano profile appears on the tangentialplane. The Fano parameter at this time is q = cot δ q (Figure 13). What does δ q mean? As shown in the cylinder’s tangential plane, δ q represents therotation of the viewing angle. This rotation can be found in the previous coupledoscillator model as a sub-system. The sub-system of mass 1 and the springs on bothsides (called system 1), which has a wide resonance due to the damping, imposes aphase shift of δ q to the other sub-system of mass 2 and the springs on both sides (calledsystem 2). he quantum and classical Fano parameter q δ =0 δ = π /2 δ = π * viewpointsin ( δ ) on cylindrical coordinateLorentzian on tangential coordinate Figure 12.
Lorentzian on tangential plane of cylindrical coordinate δ =0 δ = π /2 δ = π δ =3 π /2 * viewpointsin ( δ ) on cylindrical coordinateFano profile q=1 on tangential coordinate Figure 13.
Fano profile ( q = 1) on tangential plane of cylindrical coordinate The resonance of system 2 appears in the value | c ( ω ) | , which is proportional tothe total energy of system 1. In other words, we see the resonance of system 2 as theeffect manifests in system 1. The validity of this interpretation depends on the phasedifference of system 1 at the particular force frequency. If the phase difference of system1 is 0 or π we cannot see an asymmetric peak due to system 2. Additionally if we seethe resonance of system 1 through the total energy of system 2 ( ∝ | c ( ω ) | ) we cannotsee any asymmetric peak because γ = 0 (the phase difference of system 2 is only 0 and π except natural frequency).It does not have to be a coupled oscillator as long as this interplay of the shift ofthe phase difference occurs. Taking the two-electron excitation of helium as an example,the Fano shape occurs because the ionizing states (continuum states) cause a shift of he quantum and classical Fano parameter q
4. Conclusion
The Fano resonance is an ubiquitous phenomenon that can exist in many areas ofphysics and related fields. Therefore, the appearance of asymmetric peaks where thereis a minimum point going down to zero could have been known since long ago. One ofthe important aspects of Fano’s theory was that it provided the condition that led tothe phenomenon in a quantum system and gave a concise expression to the peak shapedefined as the Fano profile. Previous work on classical Fano resonances emphasizedonly the visual apparance, without actual derivation and interpretation of the Fanoparameter q .The definition of the Fano parameter in the quantum system in Fano (1961) isdescribed as q = (cid:104) φ | T | i (cid:105) + P V (cid:82) dE c (cid:104) φ | H | ψ Ec (cid:105)(cid:104) ψ Ec | T | i (cid:105) E − E c π (cid:104) φ | H | ψ E (cid:105) (cid:104) ψ E | T | i (cid:105) (54)where | i (cid:105) is an initial state, | φ (cid:105) is a target discrete state with Energy E and | ψ E (cid:105) is atarget continuum state with Energy E . Can you understand this intuitively?Here we rigorously derive the Fano Profile for the coupled oscillator and the (singlespring) harmonic oscillator and provide physical meaning of the Fano parameter q interms of the three pictures.The first picture is the coupled oscillator model as the system to superpose adiscrete state ( γ = 0) to a continuum state. We can adjust the rate of superpositionby γ , that cause the Fano parameter varying. This is the most rough sketch of theinterpretation of Fano parameter. The second is a study from the interpretation thatthe damping coefficient γ indicates the energy dissipation rate of the system to the othersystems. When the energy dissipation rate is close to 0, that is, when q is infinite, thepeak becomes Lorentzian. As the energy dissipation rate increases, the peak becomesasymmetric, and as the energy dissipation rate approaches infinity the peak becomes aninverted Lorentzian like a dip, called the window resonance. That is, the Fano parameter q can be regarded as a value proportional to the reciprocal of energy dissipation rate.The previous two pictures are based on the coupled oscillator system. Howeverthe foundation of the Fano resonance is not only on the coupled oscillator system orthe related systems. The most essential picture is the one-spring model through thephase difference between the external force and the displacement of the mass. Whenthe phase difference is artificially changed by δ q , it becomes increasingly asymmetricby q = − cot δ q . It is actually impossible to artificially change the phase difference.However we can find it as a sub-system of the coupled oscillator. he quantum and classical Fano parameter q q describe the gap of the each phase difference. Weviewed three pictures but it is just an “equivalent interpretation” simply because thewording is different. In quantum theory, it is not easy to understand the equivalencebetween the different schemes of Fano resonance (perturbation theory base / scatteringtheory base). The equivalence of these scheme becomes more transparent in the classicalsystem. Acknowledgments
This material is based on work supported by the Japan Society for the Promotionof Science through Grants-in-Aid for Scientific Research (No. 17K05600). We wereindebted to the following reserchers. H. Sakama (Sophia University, Faculty of Scienceand Technology), I. Tsutsui (KEK, Institute for Particle and Nuclear Studies), H. Sugio(Sophia University, Department of Philosophy) and A. Ichimura (Institute of Space andAstronautical Science, JAXA). Thank you very much.
Appendix A. The prehistory of the Fano resonance
Appendix A.1. Krypton Auger processes and helium double photoexcitation
Normal resonances have a well-known symmetric peak-shape called the Lorentzian,Breit–Wigner, or Cauchy distribution. Beutler experimentally found in 1935 that someof the resonances in the krypton and xenon absorption spectra have asymmetrical peak-shapes on Kr Auger processes (4 p ) / nl → (4 p ) / + free electron [11]. Fano and Fermicame up with an explanation for these strange peaks—that they are the resonancesdue to the combination of one bound state and one or more continuum states. Theiridea was successful, and Fano published a paper on this a few weeks later [1] (Englishtranslation by Guido Pupillo et al. [12]) [13].Another important example, helium double photoexcitation to 2 s p P ◦ , presentedan even clearer demonstration of asymmetric peaks (Figure 1). This discovery byMadden et al. in 1965 [15] the earliest scientific result obtained by utilizing synchrotronradiation experiments. It surprised the researchers that such a clear asymmetric spectrawere obtained, and they paid particular attention not only to the position and heightof the peak but also to its shape. Their discovery was a good example of how newtheoretical developments were driven by the emergence of new experimental techniques.In 1929, Dirac declared “The underlying physical laws necessary for the mathematicaltheory of a large part of physics and the whole of chemistry are thus completelyknown” [16], but atomic physics was not finished with discovering new phenomena.Instead, many new ideas entered the stage of atomic physics after that declaration wasmade. Thus, sustainable development of new research through the productive interplay he quantum and classical Fano parameter q Appendix A.2. Earlier theoretical discoveries of related phenomena
Some related studies preceded the work of Fano (and Fermi). Breit and Wignerdiscovered an effect similar to the Fano resonance in neutrons and published their resultsin 1936 [17]. It is possible that Fano’s work was inspired by their work. In the paper byBreit et al. (1936) [17], Dirac (1927) [18] is credited with finding the earliest approachfor analyzing a superposition state consisting of discrete and continuum states. Thefirst theoretical application was by Rice (1929) [19] on molecular rotational levels. Infact, a formula similar to Fano’s was derived by Dirac (1927) as equation (18) [18].This means that Fano was not the only pioneer for this type of analyses. However, themost important practical characteristic, that the spectra should be asymmetric, was notpointed out in Dirac (1927), Rice (1929), or Breit (1936). Although in some respect,Fano’s (1935) contribution appears to be only a variation of the previous studies, hisideas were revolutionary for quantum resonances in terms of peak shape analysis andbuilding a theory accessible to and testable by both experimentalists and theoreticians.Today, at the beginning of the 21st century, various scientific fields, such as solid-statephysics and nanomaterial science, continue to benefit from our understanding of Fanoresonances.
Appendix A.3. Fano (1961)
The first paper to derive the Fano profile, Fano (1935) [1] (English translation [12]),was written in Italian. Therefore, the later paper written in English, Fano (1961) [2],became better known. (There are some important differences not relevant to the currentdiscussion.) The Fano (1961) paper achieved high recognition: as of June 2003 [14], itwas 8th in citation ranking in Physical Review, in particular, ranking first for atomicand molecular physics. Fano (1961), having a big impact on physics and chemistry, wasvery rich in content for a relatively short 13-page paper.
Appendix B. Examples of classical Fano resonance
The fact that Fano resonance can occur in the system of spring and mass meansthat Fano resonance can be found in various places in everywhere and everyday life.Immediately imaginable one is the Fano resonance using the RLC resonance circuit[6] [7] [8]. For every articles cited as references on the left, they simply pointed outthat as a result of plotting the voltage gain against the frequency of the input voltagefor the LCR circuit equivalent to the coupled oscillator, the peak becomes asymmetricand they assert that it is a metaphor of Fano resonance. Given the series of results inour paper, these are not ”metaphor”, but it can be said that Fano resonance is actuallyoccurring because Fano profile can be derived analytically. he quantum and classical Fano parameter q ω ( −∞ < ω < ∞ ) with the real part of the frequency response on thehorizontal axis (In our discussion, it is the square of the amplitude of the mass 1) andthe imaginary part of the frequency response on the vertical axis (Energy dissipationrate is expressed by it). There is a paper that plots the Nyquist diagram while changingthe Fano parameter in the system where Fano resonance actually occurs [26].Looking at examples of various Fano resonances, we notice an interesting differencethat energy loss is represented by the real part of the impedance in the AC circuit,whereas in quantum mechanics it is represented by the imaginary part of the stateenergy. As we have seen, the imaginary part of the amplitude also controls energydissipation in classical mechanics. Energy dissipation is indispensable for Fanoresonance, but the representation differs between (classical and quantum) dynamics andelectronic circuit regarding real and imaginary. Even if you invert real and imaginary youcan build the same theory, so this does not indicate something physical facts. Dependingon what you first focused on in building the theories, the real and imaginary would havebeen decided. The electronic circuit first focused on resistance. So in the theory thecontrol of energy dissipation should be the real part. References [1] Fano U 1935
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