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Physical Review A

Stationarity coefficients and short-time deviations from exponential decay in atomic resonance states

Theodoros Mercouris, Cleanthes A. Nicolaides

Published Dec 14, 2001• 17 citations
Publisher

Abstract

By solving rigorously and accurately the time-dependent Schr\"odinger equation, we have obtained numerical results for the decay probability, $P(t),$ of real, multiparticle systems, in the time domain of $t\ensuremath{\approx}0.$ Three different types of atomic nonstationary states were examined, the ${\mathrm{He}}^{\ensuremath{-}}{1s2p}^{2}{}^{4}P,$ the Ca KLM $3d5p{}^{3}{F}^{o},$ and the ${\mathrm{He}}^{\mathrm{\ensuremath{-}}}$ $1s2s2p{}^{4}{P}_{5/2},$ the last one being metastable and decaying via spin-spin interactions. The main results are that there is a ${t}^{2}$ dependence of $P(t\ensuremath{\approx}0)$ and that a time-dependent short-time decay rate can be calculated. The computed coefficients of the ${t}^{2}$ term reflect the degree of stability of the state, (i.e., the degree of proximity to the notion of the standard stationary state of quantum mechanics), and are named the stationarity coefficients. These, together with the conventional quantity of the lifetime, corresponding to the exponential decay regime, constitute intrinsic properties of each real unstable state. For the herein studied metastable state the onset of exponential decay occurs after about $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\mathrm{s},$ i.e., after a duration which is achievable experimentally with laser pulses.

DOI
10.1103/physreva.65.012112

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By solving rigorously and accurately the time-dependent Schr"odinger equation, we have obtained numerical results for the decay probability, P(t),P(t),P(t), of real, multiparticle systems, in the time domain of t\ensuremath≈0.t\ensuremath{\approx}0.t\ensuremath≈0. Three different types of atomic nonstationary states were examined, the He\ensuremath−1s2p24P,{\mathrm{He}}^{\ensuremath{-}}{1s2p}^{2}{}^{4}P,He\ensuremath−1s2p24P, the Ca KLM 3d5p3Fo,3d5p{}^{3}{F}^{o},3d5p3Fo, and the He\ensuremath−{\mathrm{He}}^{\mathrm{\ensuremath{-}}}He\ensuremath− 1s2s2p4P5/2,1s2s2p{}^{4}{P}_{5/2},1s2s2p4P5/2​, the last one being metastable and decaying via spin-spin interactions. The main results are that there is a t2{t}^{2}t2 dependence of P(t\ensuremath≈0)P(t\ensuremath{\approx}0)P(t\ensuremath≈0) and that a time-dependent short-time decay rate can be calculated. The computed coefficients of the t2{t}^{2}t2 term reflect the degree of stability of the state, (i.e., the degree of proximity to the notion of the standard stationary state of quantum mechanics), and are named the stationarity coefficients. These, together with the conventional quantity of the lifetime, corresponding to the exponential decay regime, constitute intrinsic properties of each real unstable state. For the herein studied metastable state the onset of exponential decay occurs after about 5\ifmmode×\else\texttimes\fi10\ensuremath−14s,5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\mathrm{s},5\ifmmode×\else\texttimes\fi10\ensuremath−14s, i.e., after a duration which is achievable experimentally with laser pulses.
For the understanding of irreversibility at the quantum level, the formation and decay of transient (unstable) states play a fundamental role. If the system is treated within Hermitian quantum mechanics, the resulting energy distribution of the resonance state, whose Fourier transform yields the time-dependent probability of decay, P(t),P(t),P(t), is real. The physical constraint of the lower bound in the energy spectrum introduces ``memory,'' and causes nonexponential decay (NED) to set in after $t\ensuremath{\gg}\ensuremath{\tau},$ where \ensuremath{\tau} is the lifetime defined by exponential decay. The closer to threshold the decaying state is, the earlier NED appears. Apart from the constraint of E> 0,E>~0,E> 0, the constraint of t> 0t>~0t> 0 must be accounted for at the same time. It results from the discontinuity at t=0t=0t=0 of the solution of the time-dependent Schr\"odinger equation, which breaks the unitarity associated with the S matrix and gives rise to a complex energy distribution, as a manifestation of the non-Hermitian property of the decaying states. For a narrow isolated resonance state, for which the self-energy is essentially energy-independent, analytic results for PNED(t){P}_{\mathrm{NED}}(t)PNED​(t) obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints E>0E>0E>0 and t>0t>0t>0 are taken into account. As an example of the difference in the magnitude of the PNED(t){P}_{\mathrm{NED}}(t)PNED​(t) when using a real and a complex energy distribution, application is made to the decay of the unstable ${\mathrm{He}}_{2}^{2+}{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ ground molecular state.
The exponential decay of an unstable quantum state, as described by standard Markovian theories such as Fermi's Golden Rule, assumes a simple, structureless environment. However, in complex environments characterized by disorder, long-range interactions, or strong fluctuations, local decay rates fluctuate, leading to non-Markovian dynamics and power-law ``long-time tails.'' In this paper, we apply the recently proposed \textit{hyperstatistics} framework to the problem of quantum decay in such complex environments. By considering a γγγ-distribution of local decay rates across mesoscopic domains, we derive a macroscopic survival probability governed by a qqq-exponential function. We then use the qqq-generalized Gamma function, defined via the Mellin transform of the qqq-exponential, to calculate the moments of the decay-time distribution. We show that the mean quantum lifetime is ⟨t⟩=∫0∞P(t) dt=[⟨Γ⟩(2−q)]−1,\langle t\rangle = \int_0^\infty P(t)\,dt= [\langleΓ\rangle(2-q)]^{-1},⟨t⟩=∫0∞​P(t)dt=[⟨Γ⟩(2−q)]−1, and is therefore finite for q<2q<2q<2. The convergence of the second moment, and hence of the lifetime variance, instead requires the stricter condition q<3/2q<3/2q<3/2. These results provide a refined physical interpretation: for 1<q<3/21<q<3/21<q<3/2 both the mean lifetime and its variance are finite; for 3/2≤q<23/2\le q<23/2≤q<2 the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for q≥2q\ge2q≥2 the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.
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=0t=0t=0 are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters T1{T}_{1}T1​ and T2{T}_{2}T2​ separating the three domains. The parameter T1{T}_{1}T1​ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while T2{T}_{2}T2​ 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.
Abstract In recent years, an increasing number of experimental results have been published for a wide variety of systems in which high-precision, ultra-fast, and ultra-short laser pulses techniques have been used to study time-resolved dynamical processes. In the absence of a theoretical prediction of the time distribution of the transition probabilities, the lifetimes τ are generally extracted by fitting the decay time with an exponential, bi-exponential or three-exponential function. In fitting the data, the short-time behavior is generally neglected. The purpose of this study is to show that an explicit formula for the time distribution of the transition probability can be determined rigorously using time-dependent perturbation theory. A formula that perfectly fits an important subset of the experimental results from t = 0 to t = ∞ . We show that by following a route different from the usual procedure for deriving transition amplitudes and Einstein coefficients in time-dependent perturbation theory, one ends up with a time-dependent factor, that is, a temporal distribution function of the transition probabilities, and for the Einstein coefficients. The time distribution reported here looks in the time domain similar to the Planck distribution in the frequency domain. In fact, the behavior of the time distribution, with a maximum at 2 τ ln ⁡ 2 , and with τ playing the role of temperature, resembles the behavior of the Planck frequency distribution. We present several examples to demonstrate that the theoretical formula allows for easier fitting of the experimental results reported in the literature. We also show that the time distribution explains the difference in resonance heights between the experimental and theoretical blue laser spectra.
In this work, we introduce an information-theoretic approach for considering changes in dynamics of finitely dimensional open quantum systems governed by master equations. This experimentally motivated approach arises from considering how the averages of system observables change with time and quantifies how non-stationary the system is during a given time regime. By drawing an analogy with the exponential decay, we are able to further investigate regimes when such changes are negligible according to the logarithmic scale of time, and thus the system is approximately stationary. While this is always the case within the initial and final regimes of the dynamics, with the system respectively approximated by its initial and asymptotic states, we show that a distinct regime of approximate stationarity may arise. In turn, we establish a quantitative description of the phenomenon of metastability in open quantum systems. The initial relaxation occurring before the corresponding metastable regime and of the long-time dynamics taking place afterwards are also characterised. Furthermore, we explain how metastability relates to the separation in the real part of the master equation spectrum and connect our approach to the spectral theory of metastability, clarifying when the latter follows. All of our general results directly translate to Markovian dynamics of classical stochastic systems.