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arXiv:quant-ph/0305064 · Physical review. B, Condensed matter

Fano resonances in the overlapping regime

Alexandr I. Magunov, I. Rotter, С. И. Страхова

Published Dec 8, 2003• 45 citations
arXivHTMLPDF

Abstract

The line shape of resonances in the overlapping regime is studied by using the eigenvalues and eigenfunctions of the effective Hamiltonian of an open quantum system. A generalized expression ${q}_{k}(E)$ for the Fano parameter of the resonance state k is derived that contains the interaction of the state k with neighbored states $l\ensuremath{\ne}k$ via the continuum. It is energy dependent since the coupling coefficients between the state k and the continuum show a resonancelike behavior at the energies of the neighbored states $l\ensuremath{\ne}k.$ Under certain conditions, the energy dependent ${q}_{k}(E)$ are equivalent to the generalized complex energy independent Fano parameters that are introduced by Kobayashi et al. in analyzing experimental data. Long-lived states appear mostly isolated from one another in the cross section, also when they are overlapped by short-lived resonance states. The ${q}_{k}(E)$ of narrow resonances allow therefore to study the complicated interplay between different time scales in the regime of overlapping resonance states by controlling them as a function of an external parameter.

DOI
10.1103/physrevb.68.245305

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The line shape of resonances in the overlapping regime is studied by using the eigenvalues and eigenfunctions of the effective Hamiltonian of an open quantum system. A generalized expression qk(E){q}_{k}(E)qk​(E) for the Fano parameter of the resonance state k is derived that contains the interaction of the state k with neighbored states l\ensuremath≠kl\ensuremath{\ne}kl\ensuremath=k via the continuum. It is energy dependent since the coupling coefficients between the state k and the continuum show a resonancelike behavior at the energies of the neighbored states l\ensuremath≠k.l\ensuremath{\ne}k.l\ensuremath=k. Under certain conditions, the energy dependent qk(E){q}_{k}(E)qk​(E) are equivalent to the generalized complex energy independent Fano parameters that are introduced by Kobayashi et al. in analyzing experimental data. Long-lived states appear mostly isolated from one another in the cross section, also when they are overlapped by short-lived resonance states. The qk(E){q}_{k}(E)qk​(E) of narrow resonances allow therefore to study the complicated interplay between different time scales in the regime of overlapping resonance states by controlling them as a function of an external parameter.
Fano resonance is a well-celebrated weak-coupling phenomenon whereby a small interaction between a discrete and a continuum of excited states leads to an interference between the respective two transition amplitudes, resulting in an asymmetric line shape in the excitation spectra. We present the strong-coupling phase of the Fano's model where the interaction is localized in a region $\ensuremath{\xi}$ that is smaller than the collective coupling between the discrete state and the continuum. We explore the behavior of the excitation probability $\ensuremath{\sigma}$ as a function of $\ensuremath{\xi}$, the ratio of the bare transition amplitudes, and the energy of the discrete state. In the strong-coupling phase, the spectral function of the discrete state splits up and produces up to three complete destructive interferences or zeros of $\ensuremath{\sigma}$ along with ultrasharp features. We also find that the discrete state does not have to lie within the continuum to produce the well-known weak-coupling profiles of $\ensuremath{\sigma}$ as long as the collective coupling is strong enough. In addition, we consider a broadened state instead of a continuum and present a method that exactly treats such cases by explicitly including the source of the broadening---a weak coupling to the continuum of environment states---in the model.
This paper investigates the Fano-Feshbach resonance with a two-channel coupled-square-well model in both the frequency and time domains. This system is shown to exhibit Fano line-shape profiles in the energy absorption spectrum. The associated time-dependent transition moment response has a phase shift that has recently been understood to be related to the Fano line-shape asymmetric qqq parameter by $\ensuremath{\varphi}=2arg(q\ensuremath{-}i)$. The present study demonstrates that the phase-qqq correspondence is general for any Fano resonance in the weak-coupling regime, independent of the transition mechanism.
In this thesis, we study the dynamics of open quantum systems in the overlapping resonances regime. An open quantum system consists of a small, finite-dimensional system S and a reservoir R, which interacts with the system S via some operator. The reservoir is, for example, modeled by a spatially infinitely extended free Bose gas in thermal equilibrium at positive temperature. One of the main tasks in the theory of open quantum system is to examine quantum phenomena observed in the small system, such as return to equilibrium and decoherence, brought about by the interaction with the environment. To deal with such a problem, one tries to study the dynamics of the total system S + R, which is generated by the total Hamiltonian. By tracing out the degrees of freedom of the reservoir, we obtain the dynamics of the small system. Over the last decade and a half, a perturbation theory based on quantum resonance methods has been developed in the isolated resonances regime. The perturbation theory developed so far permits a mathematically rigorous treatment of the dynamics for fixed, small system-reservoir coupling parameter λ. However, in complex quantum systems, e.g. when the dimension of small system is large, the problem belongs to the overlapping resonances regime. The theory mentioned above is not applicable in this regime. In this thesis, we adapt the perturbation theory for the treatment of such a regime. We first obtain a representation formula of the reduced dynamics involving resonances of an associated operator. By analyzing the behaviour of resonances, we derive the reduced density matrix of the small system. Furthermore, we consider the spin-boson model and analyze the transition from the isolated resonances regime to the overlapping one by using the Feshbach map. We also find a critical value, related to the parameters, which separates the two regimes, marking a sharp transition in the behaviour of the resonance energies.
While Efimov physics in ultracold atoms is usually modeled with an isolated Feshbach resonance, many real world resonances appear in close vicinity to each other and are therefore overlapping. Here we derive a realistic model based on the mutual coupling of an open channel and two closed molecular channels while neglecting short-range physics as permitted by the narrow character of the considered resonances. The model is applied to three distinct scenarios with experimental relevance. We show that the effect of overlapping resonances is manifested most strikingly at a narrow resonance in whose vicinity there is a slightly narrower one. In this system the Efimov ground state extends not only over the scattering length zero crossing between the two resonances but also over the pole of the second resonance to finally meet the dissociation threshold below it. In the opposite scenario, when a narrow resonance is considered in the vicinity of a slightly broader one, we observe that the Efimov features are pushed to lower binding energies and smaller scattering lengths by a significant factor facilitating their experimental investigation. Both scenarios are compared with the case of two narrow resonances which are far enough away from each other to be effectively decoupled. In this case the two-channel model results are recovered. Finally, we analyze the rich excitation spectrum of the system and construct and explain its nodal pattern.
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.
A relation between the eigenvalues of an effective Hamilton operator and the poles of the S matrix is derived that holds for isolated as well as for overlapping resonance states. The system may be a many-particle quantum system with two-body forces between the constituents or it may be a quantum billiard without any two-body forces. Avoided crossings of discrete states as well as of resonance states are traced back to the existence of branch points in the complex plane. Under certain conditions, these branch points appear as double poles of the S matrix. They influence the dynamics of open as well as of closed quantum systems. The dynamics of the two-level system is studied in detail analytically as well as numerically.