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

arXiv:0709.2685 · Physical Review A

Long-time deviations from exponential decay for inverse-square potentials

Joan Martorell, Juan Gonzalo Muga, Donald W. L. Sprung

Published Apr 22, 2008• 21 citations
arXivHTMLPDF

Abstract

Quantal systems are predicted to show a changeover from exponential to a slower decay rate at very long times. Asymptotically most models predict power-law decay with integer exponents. However, the postexponential decay of a trapped particle from a potential can occur with a continuous range of power-law exponents. We show that this happens when the outer part of the potential is repulsive and decreases with the inverse square of the distance. We demonstrate a simple relation between the strength of the long-range tail and the power-law exponent. We also give explicit forms for the postexponential decay laws when the outer part of the potential is weakly attractive. These systems are amenable to experimental scrutiny.

DOI
10.1103/physreva.77.042719

Related papers

Nonperturbative Analysis of a Quantum Mechanical Model for Unstable Particles

U. Aglietti, Paolo Maria Santini

We present a detailed non-perturbative analysis of the time-evolution of a well-known quantum-mechanical system - a particle between potential walls - describing the decay of unstable states. For sufficiently high barriers, corresponding to unstable particles with large lifetimes, we find an exponential decay for intermediate times, turning into an asymptotic power decay. We explicitly compute such power terms in time as a function of the coupling in the model. The same behavior is obtained with a repulsive as well as with an attractive potential, the latter case not being related to any tunnelling effect.

arXiv:1010.5926•Oct 28, 2010•5 citations

Long-time behavior of many-particle quantum decay

Adolfo del Campo

While exponential decay is ubiquitous in nature, deviations at both short and long times are dictated by quantum mechanics. Nonexponential decay is known to arise due to the possibility of reconstructing the initial state from the decaying products. We discuss the quantum decay dynamics by tunneling of a many-particle system, characterizing the long-time nonexponential behavior of the nonescape and survival probabilities. The effects of contact interactions and quantum statistics are described. It is found that, whereas for noninteracting bosons the long-time decay follows a power law with an exponent linear in the number of particles $N$, the exponent becomes quadratic in $N$ in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.

arXiv:1104.4318•Physical Review A•Jul 22, 2011•81 citations

Analysis of the Breakdown of Exponential Decays of Resonances

Adam Wyrzykowski

In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schrödinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.

arXiv:1801.02350•Acta Physica Polonica B•Jan 1, 2020•5 citations

Trapped-ion coherence does not decay purely exponentially :- a weak power-law tail is always present, and we quantify it with α ≈ 0.1–0.3

khushboo rani

Abstract This work presents an empirical characterization of long-time coherence decay in a trapped-ion qubit on the IonQ Forte-1 quantum processor using high-depth CPMG dynamical-decoupling sequences. Across 97 independent runs, the measured coherence envelope exhibits a shallow long-time curvature that a standard single-exponential model cannot capture. To summarize the behavior over a wide time range, we introduce an extended empirical envelope of the form C(t) = A exp(−t/τ) t⁻ᵅ , which significantly improves fit quality (AIC/BIC, RMSE, and residual autocorrelation) relative to a pure exponential decay. The extracted power-law exponents fall in the range 0.1 ≤ α ≤ 0.3 , indicating persistent long-time structure in the coherence decay. This extended envelope provides a compact descriptive model for benchmarking trapped-ion systems, evaluating dynamical-decoupling performance, and informing hardware-aware noise characterization. The empirical crossover timescale tₚ = ατ offers a simple indicator of when the power-law component becomes relevant within the experimental window. The methods include AWS Braket execution, IonQ device calibration reporting, nonlinear least-squares fitting, and statistical diagnostics. This dataset and analysis serve as a practical reference for long-time coherence studies in trapped-ion quantum hardware.

•Nov 20, 2025•0 citations

A simple model of a decaying quantum mechanical state

A Ludviksson

A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.

Journal of Physics A Mathematical and General•Oct 1, 1987•30 citations

The true quantum face of the “exponential” decay: Unstable systems in rest and in motion

K. Urbanowski

Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P0(t) has the exponential form starting from times much smaller than the lifetime τ up to times t ⪢τ and that P0(t) exhibits inverse power–law behavior at the late time region for times longer than the so–called crossover time T ⪢ τ (The crossover time T is the time when the late time deviations of P0(t) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P0(t) can not take the pure exponential form at any time interval including times smaller than the lifetime τ or of the order of τ and it has has an oscillating form. We also study the survival probability of moving relativistic unstable particles with definite momentum . These studies show that late time deviations of the survival probability of these particles from the exponential–like form of the decay law, that is the transition times region between exponential–like and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard considerations.

EPJ Web of Conferences•Jan 1, 2017•0 citations
Quantal systems are predicted to show a changeover from exponential to a slower decay rate at very long times. Asymptotically most models predict power-law decay with integer exponents. However, the postexponential decay of a trapped particle from a potential can occur with a continuous range of power-law exponents. We show that this happens when the outer part of the potential is repulsive and decreases with the inverse square of the distance. We demonstrate a simple relation between the strength of the long-range tail and the power-law exponent. We also give explicit forms for the postexponential decay laws when the outer part of the potential is weakly attractive. These systems are amenable to experimental scrutiny.
We present a detailed non-perturbative analysis of the time-evolution of a well-known quantum-mechanical system - a particle between potential walls - describing the decay of unstable states. For sufficiently high barriers, corresponding to unstable particles with large lifetimes, we find an exponential decay for intermediate times, turning into an asymptotic power decay. We explicitly compute such power terms in time as a function of the coupling in the model. The same behavior is obtained with a repulsive as well as with an attractive potential, the latter case not being related to any tunnelling effect.
While exponential decay is ubiquitous in nature, deviations at both short and long times are dictated by quantum mechanics. Nonexponential decay is known to arise due to the possibility of reconstructing the initial state from the decaying products. We discuss the quantum decay dynamics by tunneling of a many-particle system, characterizing the long-time nonexponential behavior of the nonescape and survival probabilities. The effects of contact interactions and quantum statistics are described. It is found that, whereas for noninteracting bosons the long-time decay follows a power law with an exponent linear in the number of particles NNN, the exponent becomes quadratic in NNN in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.
In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schrödinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.
Abstract This work presents an empirical characterization of long-time coherence decay in a trapped-ion qubit on the IonQ Forte-1 quantum processor using high-depth CPMG dynamical-decoupling sequences. Across 97 independent runs, the measured coherence envelope exhibits a shallow long-time curvature that a standard single-exponential model cannot capture. To summarize the behavior over a wide time range, we introduce an extended empirical envelope of the form C(t) = A exp(−t/τ) t⁻ᵅ , which significantly improves fit quality (AIC/BIC, RMSE, and residual autocorrelation) relative to a pure exponential decay. The extracted power-law exponents fall in the range 0.1 ≤ α ≤ 0.3 , indicating persistent long-time structure in the coherence decay. This extended envelope provides a compact descriptive model for benchmarking trapped-ion systems, evaluating dynamical-decoupling performance, and informing hardware-aware noise characterization. The empirical crossover timescale tₚ = ατ offers a simple indicator of when the power-law component becomes relevant within the experimental window. The methods include AWS Braket execution, IonQ device calibration reporting, nonlinear least-squares fitting, and statistical diagnostics. This dataset and analysis serve as a practical reference for long-time coherence studies in trapped-ion quantum hardware.
A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.
Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P0(t) has the exponential form starting from times much smaller than the lifetime τ up to times t ⪢τ and that P0(t) exhibits inverse power–law behavior at the late time region for times longer than the so–called crossover time T ⪢ τ (The crossover time T is the time when the late time deviations of P0(t) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P0(t) can not take the pure exponential form at any time interval including times smaller than the lifetime τ or of the order of τ and it has has an oscillating form. We also study the survival probability of moving relativistic unstable particles with definite momentum . These studies show that late time deviations of the survival probability of these particles from the exponential–like form of the decay law, that is the transition times region between exponential–like and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard considerations.