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

Il Nuovo Cimento B

Exponential and non-exponential decay of an initially localized quantum state subject to two delta-barriers

Mark R. A. Shegelski, Rebecca A. Booth

Published Mar 1, 2005• 0 citations
Publisher

Abstract

We study the time dependence of a quantum particle initially localized'behind two delta-function barriers of strengths v 1 and v 2 . We determine the conditions under which the decay is exponential by varying the ratio v 1 /v 2 and the separation between the delta-barriers. We find that exponential decay occurs even when v 1 /v 2 is of order one. We also find cases of non-exponential decay. An interesting example of non-exponential decay is when the two barriers are very close. We show that the exponential decay is due to poles in the complex k-plane, where k denotes the wave numbers of the energy eigenstates, and we find that the decay rates are fully understood in many cases as being due to a single pole.

DOI
10.1393/ncb/i2005-10004-9

Related papers

Decay modes of two repulsively interacting bosons

Sungyun Kim, Baebang Myun, Joachim Brand

We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.

•Aug 16, 2016•17 citations

Decay modes of two repulsively interacting bosons

Sungyun Kim, Joachim Brand

We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.

arXiv:1106.6109•Journal of Physics B Atomic Molecular and Optical Physics•Sep 1, 2011•15 citations

Capturing nonexponential dynamics in the presence of two decay channels

Francesco Giacosa, Przemysław Kościk, Tomasz Sowiński

The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the nonexponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through specific quantum mechanical examples which can be accurately solved, we study the general properties of a two-channel decay that apply for various unstable quantum states (including unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit), but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential-decay law.

arXiv:1912.06394•Physical Review A•Aug 11, 2020•14 citations

Survival Probability in Waveguides with Delta Barriers: A Study of Tunneling and Resonances

Wiktor W. Wolak, João G. M. Silva, Sylwia Kondej +1 more

We study resonances and time decay in a quantum waveguide with two semitransparent barriers. The model is first analyzed as a one-dimensional Schrödinger operator with two delta interactions and then extended to a two-dimensional strip, where transverse Dirichlet modes reduce the problem, channel by channel, to the one-dimensional setting. We derive explicit scattering amplitudes and characterize resonance poles by analytic continuation. In the strong-barrier regime ρ=α−1→0+, the real parts of the resonances approach the eigenvalues of the decoupled cavity, while the imaginary parts are negative and of order α−2, so the lifetimes grow like α2. We also study the time evolution on short, intermediate, and long time scales: the survival probability is quadratic at short times, in accordance with the quantum Zeno effect; the intermediate regime is resonance-dominated and approximately exponential; and the long-time behavior is governed by the low-energy threshold and becomes algebraic. Numerical simulations complement the analysis by confirming these regimes and by showing the breakdown of exponential decay. They also provide an empirical fit for the breakdown time tbr in terms of the barrier strength α and the distance d1 between the barriers.

Symmetry•Aug 27, 2026•0 citations

Decay versus survival of a localized state subjected to harmonic forcing: exact results

A. Rokhlenko, Ovidiu Costin, Joel L. Lebowitz

We investigate the survival probability of a localized 1D quantum particle subjected to a time-dependent potential of the form rU ( x ) sin ω t with U ( x ) = 2δ( x − a ) or U ( x ) = 2δ( x − a ) − 2δ( x + a ). The particle is initially in a bound state produced by the binding potential −2δ( x ). We prove that this probability goes to zero as t → ∞ for almost all values of r , ω and a . The decay is initially exponential followed by a t −3 law if ω is not close to resonances and r is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters r , ω and a the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behaviour even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.

arXiv:math-ph/0608031•Journal of Physics A Mathematical and General•Oct 8, 2002•18 citations

Tunneling-Induced Decay of Wave Packets Localized Beyond an Asymmetric Potential Barrier

M.L. Strekalov

The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.

Physical Science International Journal•Jun 20, 2020•0 citations
We study the time dependence of a quantum particle initially localized'behind two delta-function barriers of strengths v 1 and v 2 . We determine the conditions under which the decay is exponential by varying the ratio v 1 /v 2 and the separation between the delta-barriers. We find that exponential decay occurs even when v 1 /v 2 is of order one. We also find cases of non-exponential decay. An interesting example of non-exponential decay is when the two barriers are very close. We show that the exponential decay is due to poles in the complex k-plane, where k denotes the wave numbers of the energy eigenstates, and we find that the decay rates are fully understood in many cases as being due to a single pole.
We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.
We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.
The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the nonexponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through specific quantum mechanical examples which can be accurately solved, we study the general properties of a two-channel decay that apply for various unstable quantum states (including unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit), but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential-decay law.
We study resonances and time decay in a quantum waveguide with two semitransparent barriers. The model is first analyzed as a one-dimensional Schrödinger operator with two delta interactions and then extended to a two-dimensional strip, where transverse Dirichlet modes reduce the problem, channel by channel, to the one-dimensional setting. We derive explicit scattering amplitudes and characterize resonance poles by analytic continuation. In the strong-barrier regime ρ=α−1→0+, the real parts of the resonances approach the eigenvalues of the decoupled cavity, while the imaginary parts are negative and of order α−2, so the lifetimes grow like α2. We also study the time evolution on short, intermediate, and long time scales: the survival probability is quadratic at short times, in accordance with the quantum Zeno effect; the intermediate regime is resonance-dominated and approximately exponential; and the long-time behavior is governed by the low-energy threshold and becomes algebraic. Numerical simulations complement the analysis by confirming these regimes and by showing the breakdown of exponential decay. They also provide an empirical fit for the breakdown time tbr in terms of the barrier strength α and the distance d1 between the barriers.
We investigate the survival probability of a localized 1D quantum particle subjected to a time-dependent potential of the form rU ( x ) sin ω t with U ( x ) = 2δ( x − a ) or U ( x ) = 2δ( x − a ) − 2δ( x + a ). The particle is initially in a bound state produced by the binding potential −2δ( x ). We prove that this probability goes to zero as t → ∞ for almost all values of r , ω and a . The decay is initially exponential followed by a t −3 law if ω is not close to resonances and r is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters r , ω and a the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behaviour even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.