Resonance fluorescence from the two-level atom in the field of a resonant, exponentially-decaying pulse is considered when the spontaneous linewidth may be neglected. The analytical expressions for the physical spectra are given and discussed for any field strength and any pulse decay time. The approximation schemes for large and small pulse areas are presented and illustrated by numerical results. In the latter case line narrowing is obtained, while the former corresponds to the near-adiabatic case.Resonance fluorescence from the two-level atom in the field of a resonant, exponentially-decaying pulse is considered when the spontaneous linewidth may be neglected. The analytical expressions for the physical spectra are given and discussed for any field strength and any pulse decay time. The approximation schemes for large and small pulse areas are presented and illustrated by numerical results. In the latter case line narrowing is obtained, while the former corresponds to the near-adiabatic case.

Journal of Physics B Atomic and Molecular PhysicsJul 14, 19859 citations

Non-exponential decay of wavefunctions and scattering resonances

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

APSOct 1, 20010 citations

Time evolution of unstable quantum states and a resolution of Zeno's paradox

Charles B. Chiu, E. C. G. Sudarshan, Baidyanath Misra

The time evolution of quantum states for unstable particles can be conveniently divided into three domains: the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law. In this work, we reexamine several questions relating to the deviations from the simple exponential decay law. On the basis of general considerations, we demonstrate that deviations from exponential decay near $t=0$ are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters ${T}_{1}$ and ${T}_{2}$ separating the three domains. The parameter ${T}_{1}$ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while ${T}_{2}$ is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.The time evolution of quantum states for unstable particles can be conveniently divided into three domains: the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law. In this work, we reexamine several questions relating to the deviations from the simple exponential decay law. On the basis of general considerations, we demonstrate that deviations from exponential decay near $t=0$ are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters ${T}_{1}$ and ${T}_{2}$ separating the three domains. The parameter ${T}_{1}$ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while ${T}_{2}$ is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJul 15, 1977347 citations

Buildup and decay dynamics of exponentially modulated quantum waves in resonant structures

D Amador-Bartolini, Jorge Villavicencio, Roberto Romo +1 more

Abstract We derive an exact analytic solution of time-dependent Schrödinger’s equation for cutoff quantum waves with exponential modulation to explore the dynamics along the internal region of a one-dimensional potential of arbitrary shape. The modulation of the wave packet is controlled by a single parameter given by its momentum distribution width, Δ, which allows to explore on the same footing the transient features of both cutoff spatially localized wave packets and extended plane waves. From the formal solution, we derive a simple and reliable formula that accurately describes both the buildup and decay of the probability density inside a resonant structure. The buildup-decay formula involves two exponential terms governed respectively by the resonance width Γ 1 of the lowest resonance of the system and the width Γ ¯ of a ‘virtual’ state induced by the incident wave packet. During the buildup process, we demonstrate that the contributions of both Γ 1 and Γ ¯ become essential. In the decay process, on the other hand, we find that there are two regimes characterized by a critical value Δ c , such that for Δ > Δ c the exponential decay is governed by Γ 1 , while for Δ < Δ c the exponential decay is governed by Γ ¯ . We also derive an analytic formula for a relevant timescale for the buildup-decay process, which depends in a simple fashion on the decay rates Γ 1 and Γ ¯ . Deviations from the exponential decay law at long times are also discussed.Abstract We derive an exact analytic solution of time-dependent Schrödinger’s equation for cutoff quantum waves with exponential modulation to explore the dynamics along the internal region of a one-dimensional potential of arbitrary shape. The modulation of the wave packet is controlled by a single parameter given by its momentum distribution width, Δ, which allows to explore on the same footing the transient features of both cutoff spatially localized wave packets and extended plane waves. From the formal solution, we derive a simple and reliable formula that accurately describes both the buildup and decay of the probability density inside a resonant structure. The buildup-decay formula involves two exponential terms governed respectively by the resonance width Γ 1 of the lowest resonance of the system and the width Γ ¯ of a ‘virtual’ state induced by the incident wave packet. During the buildup process, we demonstrate that the contributions of both Γ 1 and Γ ¯ become essential. In the decay process, on the other hand, we find that there are two regimes characterized by a critical value Δ c , such that for Δ > Δ c the exponential decay is governed by Γ 1 , while for Δ < Δ c the exponential decay is governed by Γ ¯ . We also derive an analytic formula for a relevant timescale for the buildup-decay process, which depends in a simple fashion on the decay rates Γ 1 and Γ ¯ . Deviations from the exponential decay law at long times are also discussed.

Physica ScriptaOct 22, 20202 citations

An explicit model of exponential decay

David P. Vasholz

PhysicaJun 1, 19743 citations

t=0

are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters

{T}_{1}

and

{T}_{2}

separating the three domains. The parameter

{T}_{1}

is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while

{T}_{2}

is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.