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Physical Review Letters

Possibility of Observing Nonexponential Decays in Autoionizing States

Cleanthes A. Nicolaides, Donald R. Beck

Published Mar 28, 1977• 39 citations
Publisher

Abstract

A rigorous evaluation of the time evolution amplitude $G(t)$ for an isolated autoionizing state has yielded the result ${{|G(t)|}^{2}}_{t\ensuremath{\rightarrow}\ensuremath{\infty}}={e}^{\ensuremath{-}\frac{\ensuremath{\Gamma}t}{\ensuremath{\hbar}}+\frac{{\ensuremath{\hbar}}^{2}}{{\ensuremath{\pi}}^{2}(4{E}^{2}+{\ensuremath{\Gamma}}^{2}){t}^{2}}}$ where $E$ is the energy above the ionization threshold and $\ensuremath{\Gamma}$ the width. For states with lifetimes in the region ${10}^{\ensuremath{-}8}$-${10}^{\ensuremath{-}11}$ sec near the ionization threshold, the nonexponential decay corrections are sufficiently large to suggest that perhaps it is possible to test the validity of the fundamental exponential "law" within the realm of atomic physics.

DOI
10.1103/physrevlett.38.683

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A rigorous evaluation of the time evolution amplitude G(t)G(t)G(t) for an isolated autoionizing state has yielded the result ∣G(t)∣2t\ensuremath→\ensuremath∞=e\ensuremath−\ensuremathΓt\ensuremathℏ+\ensuremathℏ2\ensuremathπ2(4E2+\ensuremathΓ2)t2{{|G(t)|}^{2}}_{t\ensuremath{\rightarrow}\ensuremath{\infty}}={e}^{\ensuremath{-}\frac{\ensuremath{\Gamma}t}{\ensuremath{\hbar}}+\frac{{\ensuremath{\hbar}}^{2}}{{\ensuremath{\pi}}^{2}(4{E}^{2}+{\ensuremath{\Gamma}}^{2}){t}^{2}}}∣G(t)∣2t\ensuremath→\ensuremath∞​=e\ensuremath−\ensuremathℏ\ensuremathΓt​+\ensuremathπ2(4E2+\ensuremathΓ2)t2\ensuremathℏ2​ where EEE is the energy above the ionization threshold and \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ the width. For states with lifetimes in the region 10\ensuremath−8{10}^{\ensuremath{-}8}10\ensuremath−8-10\ensuremath−11{10}^{\ensuremath{-}11}10\ensuremath−11 sec near the ionization threshold, the nonexponential decay corrections are sufficiently large to suggest that perhaps it is possible to test the validity of the fundamental exponential "law" within the realm of atomic physics.
The decay profile of a metastable quantum state decaying in isolation is predicted to depart from exponential behaviour after many lifetimes, and in typical cases to adopt algebraic behaviour as the late-time limit. The effect of environmental interactions, including measurement, is quite different from that of the familiar Zeno (and anti-Zeno) effects that have been observed at very early times. Instead of arresting the decay (or hastening it), the late-time interactions tend to enforce the exponential decay law, without substantially changing the decay constant intrinsic to the isolated system. The experimental challenge is to prevent these interactions and observe the crossover to the nonexponential regime.
The most important law of radioactivity is that of the exponential decay. In the realm of quantum mechanics, however, this decay law is neither rigorous nor fundamental. The deviations from the exponential decay have been observed experimentally at the early stage of a decay process, but there is little evidence for nonexponential behavior at long times. Yet such long-term nonexponentiality is expected theoretically to probe the nonresonant background components of the initial wave function which preserve the structural interference and the memory of how the state was created. In this paper, we propose new observables that can be used for experimental investigations of the post-exponential decay regime, including the decay of threshold resonances, particle correlations in three-body decays, and interference between near-lying resonances. While the specific examples presented in this work pertain to atomic nuclei, the properties of nonexponential decay are generic, i.e., they apply to other many-body open quantum systems, such as hadrons, atoms, molecules, and nanostructures.
Abstract Stationary‐state quantum mechanics presents no difficulties in defining and computing discrete excited states because they obey the rules established in the properties of Hilbert space. However, when this idealization has to be abandoned to formulate a theory of excited states dissipating into a continuous spectrum, the problem acquires additional interest in many fields of physics. In this article, the theory of resonances in the continuous spectrum is formulated as a problem of decaying states, whose treatment can entail time‐dependent as well as energy‐dependent theories. The author focuses on certain formal and computational issues and discusses their application to polyelectronic atomic states. It is argued that crucial to the theory is the understanding and computation of a multiparticle localized wavepacket, Ψ0, at t = 0, having a real energy E0. Assuming this as the origin, without memory of the excitation process, the author discusses aspects of time‐dependent dynamics, for t ≈ 0 as well as for t → ∞, and the possible significance of nonexponential decay in the understanding of time asymmetry. Also discussed is the origin of the complex eigenvalue Schrödinger equation (CESE) satisfied by resonance states and the state‐specific methodology for its solution. The complex eigenvalue drives the decay exponentially, with a rate Γ, to a good approximation. It is connected to E0 via analytic continuation of the complex self‐energy function, A(z), (z is complex), into the second Riemann sheet, or, via the imposition of outgoing wave boundary conditions on the stationary state Schrödinger equation satisfied by the Fano standing wave superposition in the vicinity of E0. If the nondecay amplitude, G(t), is evaluated by inserting the unit operator I = ∫dE|E><E| G(t) = < Ψ0|e−iHt|Ψ0 >, then the resulting spectral function is real, g(E) =|<Ψ0|E>|2, and does not differentiate between positive and negative times. The introduction of time asymmetry, which is associated with irreversibility, is achieved by starting from < Ψ0|θ(t)e−iHt|Ψ0 >, where θ(t) is the step function at the discontinuity point t = 0. In this case, the spectral function is complex. Within the range of validity of exponential decay, the complex spectral function is the same as the coefficient of Ψ0 in the theory of the CESE. A calculation of G(t) using the simple pole approximation and the constraints that t > 0 and E > 0 results in a nonexponential decay (NED) correction for t ≫ 1/Γ that is different than when a real g(E) is used, representing the contribution of both “in” and “out” states. Earlier formal and computational work has shown that resonance states close to threshold are good candidates for NED to acquire nonnegligible magnitude. In this context, a pump‐probe laser experiment in atomic physics is proposed, using as a paradigm the He− 1s2p2 4P shape resonance. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002