Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Optical Society of America B

Collisionally induced configuration mixing: soluble model with continuous spectrum

Martin Wilkens, Kazimierz Rza̧żewski

Published Nov 1, 1988• 0 citations
Publisher

Abstract

A model of strong bound-free coupling by a stochastic train of delta-function impulses is solved analytically. The model can be used to describe transitions caused by strong isolated collisions. The decay of bound-state population and photoelectron spectra is discussed, and the limits of the applicability of Fermi’s Golden Rule are evaluated. It turns out that the results depend nontrivially on the statistics of the driving signal.

DOI
10.1364/josab.5.002289

Related papers

Generalisation of Fermi's golden rule for the strong bound-free transitions

R J Kuklinski, Kazimierz Rza̧żewski

A simple analysis of the bound-free system shows that for a narrow continuum the Fermi golden rule can be generalised for the high-intensity regime. This generalisation has the form of a simple expression for the decay rate and helps to explain nearly perfect population trapping in the case of a narrowly peaked matrix element. Such population trapping means that the decay rate of a bound state strongly coupled to a continuum tends to zero for increasing coupling strength only if the matrix element falls off faster than the Lorentzian. The authors also find new solutions for pulsed excitation giving exact population trapping in the bound-free transition.

Journal of Physics A Mathematical and General•Oct 7, 1989•0 citations

Soluble quantum map with continuous spectrum

Kazimierz Rza̧ewski, Jan Mostowski

We present a soluble model of an atomic system driven by a train of \ensuremath{\delta}-function impulses. The atomic Hamiltonian describes an atom with one bound state and one continuum representing an autoionizing resonance. The model is soluble or requires very simple numerics. The results depend strongly on the shape of the resonance. Population trapping is discovered for a Gaussian-shaped continuum. Corrections coming from the counterrotating terms are discussed.

Physical Review A•May 1, 1987•9 citations

Continuity of Bound and Unbound States in a Fermi Gas: A Soluble Example

Chanchal K. Majumdar

Considering a gas of independent fermions in the presence of an attractive localized potential, one can show that the properties of the system as a whole are smooth (analytic) functions of the strength parameter of the potential, even at those values where new single-particle bound states appear. Thus for the system as a whole, the transition from ``unbound'' to ``bound'' states is continuous and the concept of a bound state cannot be made precise. This is illustrated here for a simple mathematically soluble model—noninteracting spinless fermions moving in the presence of a delta-function potential in one dimension. Some related physical ideas are also presented.

Journal of Mathematical Physics•Apr 1, 1966•3 citations

Fermi’s golden rule: its derivation and breakdown by an ideal model

Jian‐Min Zhang, Yueheng Liu

Abstract Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from − ∞ to + ∞ , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time , even when the first order perturbation approximation itself is still valid.

arXiv:1604.06916•European Journal of Physics•Oct 6, 2016•53 citations

The rotating-wave approximation in bound-free transitions

Juha Javanainen

The author solves the Schrodinger equation in terms of continued fractions for a system exhibiting light-induced bound-free transitions over an energy gap, without making the rotating-wave approximation. He describes the evolution of the bound state and the photoelectron spectrum by investigating the analytic properties of the Fourier transforms of the probability amplitudes. The main consequence of non-RWA transitions is a Bloch-Siegert shift of the continuum threshold, which may be comparable to RWA dynamic level shifts. At high intensities the counter-rotating wave also influences the evolution of the bound state, and gives rise to multiphoton transitions competing with RWA multiphoton transitions with an odd number of photons, but these effects are probably hard to isolate experimentally.

Journal of Physics B Atomic and Molecular Physics•Apr 28, 1983•26 citations

Emergence of Fermi’s Golden Rule

Tobias Micklitz, Alan Morningstar, Alexander Altland +1 more

Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.

arXiv:2206.00704•Physical Review Letters•Sep 27, 2022•31 citations
A model of strong bound-free coupling by a stochastic train of delta-function impulses is solved analytically. The model can be used to describe transitions caused by strong isolated collisions. The decay of bound-state population and photoelectron spectra is discussed, and the limits of the applicability of Fermi’s Golden Rule are evaluated. It turns out that the results depend nontrivially on the statistics of the driving signal.
A simple analysis of the bound-free system shows that for a narrow continuum the Fermi golden rule can be generalised for the high-intensity regime. This generalisation has the form of a simple expression for the decay rate and helps to explain nearly perfect population trapping in the case of a narrowly peaked matrix element. Such population trapping means that the decay rate of a bound state strongly coupled to a continuum tends to zero for increasing coupling strength only if the matrix element falls off faster than the Lorentzian. The authors also find new solutions for pulsed excitation giving exact population trapping in the bound-free transition.
Considering a gas of independent fermions in the presence of an attractive localized potential, one can show that the properties of the system as a whole are smooth (analytic) functions of the strength parameter of the potential, even at those values where new single-particle bound states appear. Thus for the system as a whole, the transition from ``unbound'' to ``bound'' states is continuous and the concept of a bound state cannot be made precise. This is illustrated here for a simple mathematically soluble model—noninteracting spinless fermions moving in the presence of a delta-function potential in one dimension. Some related physical ideas are also presented.
Abstract Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from − ∞ to + ∞ , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time , even when the first order perturbation approximation itself is still valid.
The author solves the Schrodinger equation in terms of continued fractions for a system exhibiting light-induced bound-free transitions over an energy gap, without making the rotating-wave approximation. He describes the evolution of the bound state and the photoelectron spectrum by investigating the analytic properties of the Fourier transforms of the probability amplitudes. The main consequence of non-RWA transitions is a Bloch-Siegert shift of the continuum threshold, which may be comparable to RWA dynamic level shifts. At high intensities the counter-rotating wave also influences the evolution of the bound state, and gives rise to multiphoton transitions competing with RWA multiphoton transitions with an odd number of photons, but these effects are probably hard to isolate experimentally.
Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.