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 Physics B Atomic Molecular and Optical Physics

Non-exponential decay due to population trapping in an infinite sequence of discrete states and continua

Edward J. Robinson

Published Jul 28, 1994• 3 citations
Publisher

Abstract

A calculation is presented of population trapping by an infinite sequence of bound states and continua, with results that differ from both the pure exponential decay exhibited by a single bound state, and from the combination of exponential decay and no decay that is obtained when there are two bound states coupled to a single continuum. Instead of either of these, the theory predicts a rule for the decay that becomes, for time t large (but not so large that events are unobservable), an inverse power law in t 1 2/. Possible experimental realizations of this problem are discussed, which also indicate that the Floquet form of the wavefunction does not apply.

DOI
10.1088/0953-4075/27/14/002

Related papers

Amplification of non‐Markovian decay due to bound state absorption into continuum

Savannah Sterling Garmon, T. Petrosky, L. Simine +1 more

Abstract It is known that quantum systems yield non‐exponential (power law) decay on long time scales, associated with continuum threshold effects contributing to the survival probability for a prepared initial state. For an open quantum system consisting of a discrete state coupled to continuum, we study the case in which a discrete bound state of the full Hamiltonian approaches the energy continuum as the system parameters are varied. We find in this case that at least two regions exist yielding qualitatively different power law decay behaviors; we term these the long time ‘near zone’ and long time ‘far zone’. In the near zone the survival probability falls off according to a t‐1 power law, and in the far zone i t falls off as t‐3. We show that the timescale TQ separating these two regions is inversely related to the gap between the discrete bound state energy and the continuum threshold. In the case that the bound state is absorbed into the continuum and vanishes, then the time scale TQ diverges and the survival probability follows the t‐1 power law even on asymptotic scales. Conversely, one could study the case of an anti‐bound state approaching the threshold before being ejected from the continuum to form a bound state. Again the t‐1 power law dominates precisely at the point of ejection.

arXiv:1204.6141•Fortschritte der Physik•Jul 5, 2012•45 citations

Population trapping in laser-induced continuum-continuum transitions

Kazimierz Rza̧żewski, Joseph W. Haus, L Wang

The authors show that the population trapping phenomenon found by Deng and Eberly (1986, 1988) for free-free transitions between structureless continua occurs under much more general conditions. They solve the time-dependent models exactly and show for asymptotically long times that population trapping can occur for two continua with energy-dependent matrix elements, and for an infinite number of structureless continua with coupled bands. Furthermore, they show that the trapping persists for two continua coupled by a noisy laser. The amount of trapping depends on the initial conditions.

Journal of Physics B Atomic Molecular and Optical Physics•Oct 28, 1989•2 citations

Continuum-continuum population trapping

Ronald S. Burkey, C. A. Glosson, Cyrus D. Cantrell

We consider the problem of coherent population trapping in a model quantum system consisting of two (and briefly three) coupled continua of energy levels. While an earlier study [Z. Deng and J. H. Eberly, Phys. Rev. A 34, 2492 (1986)] has demonstrated population trapping in a closely related system, it has done so only for the case of a broad featureless continuum. We will demonstrate, on the other hand, that the time evolution of the continuum is qualitatively very different when different types of continua are allowed, and that population trapping is not likely to occur except in the broad featureless continuum. We do show, however, that in some cases other counterintuitive results surface.

Physical Review A•Mar 1, 1989•4 citations

Quantum decay into a non-flat continuum

James Aisenberg, Itamar Sela, Tsampikos Kottos +2 more

We study the decay of a prepared state into non-flat continuum. We find that the survival probability P ( t ) might exhibit either stretched-exponential or power-law decay, depending on non-universal features of the model. Still there is a universal characteristic time t 0 that does not depend on the functional form. It is only for a flat continuum that we get a robust exponential decay that is insensitive to the nature of the intra-continuum couplings. The analysis highlights the co-existence of perturbative and non-perturbative features in the local density of states, and the nonlinear dependence of 1/ t 0 on the strength of the coupling.

arXiv:0905.4934•Journal of Physics A Mathematical and Theoretical•Feb 9, 2010•1 citation

Time-dependent pulse and frequency behavior of population trapping via the continuum

Emmanuel Paspalakis, M. Protopapas, P. L. Knight

Summary form only given. We study two fundamental systems that involve continua: a system with a single autoionizing state and the laser induced continuum structure scheme (LICS), both shown. We pay special attention to the response of these systems to time-dependent pulse interactions, which we find to differ from that of square pulse interactions. Such a study is particularly useful from the point of view of the many applications associated with these systems. For the theoretical analysis we perform appropriate transformations that allow us to study both systems simultaneously. From an analysis of the resulting Hamiltonian of these systems, we determine the important conditions for efficient population trapping.

Conference on Lasers and Electro-Optics•Nov 27, 2002•0 citations

Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times

A. Petridis, L. P. Staunton, Jon Vermedahl +1 more

The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.

Journal of Modern Physics•Jan 1, 2010•6 citations
A calculation is presented of population trapping by an infinite sequence of bound states and continua, with results that differ from both the pure exponential decay exhibited by a single bound state, and from the combination of exponential decay and no decay that is obtained when there are two bound states coupled to a single continuum. Instead of either of these, the theory predicts a rule for the decay that becomes, for time t large (but not so large that events are unobservable), an inverse power law in t 1 2/. Possible experimental realizations of this problem are discussed, which also indicate that the Floquet form of the wavefunction does not apply.
Abstract It is known that quantum systems yield non‐exponential (power law) decay on long time scales, associated with continuum threshold effects contributing to the survival probability for a prepared initial state. For an open quantum system consisting of a discrete state coupled to continuum, we study the case in which a discrete bound state of the full Hamiltonian approaches the energy continuum as the system parameters are varied. We find in this case that at least two regions exist yielding qualitatively different power law decay behaviors; we term these the long time ‘near zone’ and long time ‘far zone’. In the near zone the survival probability falls off according to a t‐1 power law, and in the far zone i t falls off as t‐3. We show that the timescale TQ separating these two regions is inversely related to the gap between the discrete bound state energy and the continuum threshold. In the case that the bound state is absorbed into the continuum and vanishes, then the time scale TQ diverges and the survival probability follows the t‐1 power law even on asymptotic scales. Conversely, one could study the case of an anti‐bound state approaching the threshold before being ejected from the continuum to form a bound state. Again the t‐1 power law dominates precisely at the point of ejection.
The authors show that the population trapping phenomenon found by Deng and Eberly (1986, 1988) for free-free transitions between structureless continua occurs under much more general conditions. They solve the time-dependent models exactly and show for asymptotically long times that population trapping can occur for two continua with energy-dependent matrix elements, and for an infinite number of structureless continua with coupled bands. Furthermore, they show that the trapping persists for two continua coupled by a noisy laser. The amount of trapping depends on the initial conditions.
We consider the problem of coherent population trapping in a model quantum system consisting of two (and briefly three) coupled continua of energy levels. While an earlier study [Z. Deng and J. H. Eberly, Phys. Rev. A 34, 2492 (1986)] has demonstrated population trapping in a closely related system, it has done so only for the case of a broad featureless continuum. We will demonstrate, on the other hand, that the time evolution of the continuum is qualitatively very different when different types of continua are allowed, and that population trapping is not likely to occur except in the broad featureless continuum. We do show, however, that in some cases other counterintuitive results surface.
We study the decay of a prepared state into non-flat continuum. We find that the survival probability P ( t ) might exhibit either stretched-exponential or power-law decay, depending on non-universal features of the model. Still there is a universal characteristic time t 0 that does not depend on the functional form. It is only for a flat continuum that we get a robust exponential decay that is insensitive to the nature of the intra-continuum couplings. The analysis highlights the co-existence of perturbative and non-perturbative features in the local density of states, and the nonlinear dependence of 1/ t 0 on the strength of the coupling.
Summary form only given. We study two fundamental systems that involve continua: a system with a single autoionizing state and the laser induced continuum structure scheme (LICS), both shown. We pay special attention to the response of these systems to time-dependent pulse interactions, which we find to differ from that of square pulse interactions. Such a study is particularly useful from the point of view of the many applications associated with these systems. For the theoretical analysis we perform appropriate transformations that allow us to study both systems simultaneously. From an analysis of the resulting Hamiltonian of these systems, we determine the important conditions for efficient population trapping.
The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.