On the violation of the exponential decay law in atomic physics:ab initiocalculation of the time-dependence of the non-stationary state

Abstract

The detailed time dependence of the decay of a three-electron autoionizing state close to threshold has been obtained ab initio by solving the time-dependent Schrödinger equation (TDSE). The theory allows the definition and computation of energy-dependent matrix elements in terms of the appropriate N-electron wavefunctions, representing the localized initial state, 0 , the stationary scattering states of the continuous spectrum, U(ε), and the localized excited states, n , of the effective Hamiltonian QH Q, where Q ≡ | 0 0 |. The time-dependent wavefunction is expanded over these states and the resulting coupled equations with timedependent coefficients (in the thousands) are solved to all orders by a Taylor series expansion technique. Convergence is checked as a function of the number of the numerically obtained U(ε) that span the continuous spectrum of the free electron. The robustness of the method was verified by using a model interaction in analytic form and comparing the results from two different methods for integrating the TDSE (appendix B). For the physically relevant application, the chosen state was the He − 1s2p 2 4 P shape resonance, about which very accurate theoretical and experimental relevant information exists. Calculations using accurate wavefunctions and an energy grid of 20.000 points in the range 0.0-21.77 eV show that the effective interaction depends on energy in a state-specific manner, thereby leading to state-specific characteristics of non-exponential decay (NED). For the established energy position of 0.01 eV, the results show an exponential decay over about 6 × 10 4 au of time, from which a width of = 5.2 meV and a lifetime of 1.26 × 10 −13 s is deduced. The experimentally obtained width is 7.16 meV (Walter, Seifert and Peterson 1994 Phys. Rev. A 50 664). After 12 lifetimes (about 1400 fs), at which time the survival probability is 10 −6 , NED sets in. On the other hand, due to the shape of the interaction, the NED appears at earlier times if the energy position happened to be slightly larger. For example, if E were at 0.019 eV, NED would start after nine exponential lifetimes. These facts suggest that either in this state or in other autoionizing states close to threshold, NED may have sufficient presence to make the violation of the law of exponential decay observable.

Violation of the Exponential-Decay Law at Long Times

“…For the radioactive isotope 60 Co, Norman et al observed the decay out to 30 exponential lifetimes, whereas theoretically the deviation from exponential decay is expected at signal intensities of order 10 ÿ60 of the initial intensity corresponding to 140 lifetimes [5]. More recently, theoretical effort has been directed to a range of atomic states including spontaneous photon emission [6], or autoand photoionizing states with very small release energies (E 0:01 eV) [7]. A common feature for all these atomic systems are the well-defined energy levels, exhibiting only homogeneous broadening, accompanied by (nearly) Lorentzian line shapes and exponential lifetimes that are inversely proportional to the width of the energetic state distribution, @=ÿ.…”

Nonperturbative computation of the time-resolved formation of the profile of autoionizing states as a function of the intensity and duration of ultrashort pulses

“…Indeed, about 15 years ago [3,4], we demonstrated the possibility of computing from first principles the time-resolved decay of prototypical unstable states of polyelectronic atoms. The computations produced energies, lifetimes, and energy distributions.…”

“…The computations produced energies, lifetimes, and energy distributions. Most importantly, in view of the well-documented "law" of exponential decay (ED), emphasis was placed on the fundamental issue of determining the magnitudes of the deviation from ED for very long [3,4] and for very short [5] times, using as test cases not simple models and formal manipulations, but real many-electron systems where the operators causing the decay are nonrelativistic or relativistic (Breit-Pauli Hamiltonian) and where the N-electron wave functions include the information of electronic structure and of electron correlation.…”

Non-exponential tunneling ionization of atoms by an intense laser field

“…The following discussions revealed a number of possible mechanisms for non-exponential decay for long as well as short times (see, e.g., ). Different physical situations have been considered starting from a basic model of penetration through a delta-edged potential [5], and including such processes as the spontaneous decay in two-level systems [6], tunneling alpha-decay of atomic nuclei and cold electron emission from metals [7], single photon ionization of atoms [8], non-exponential decays in autoionizing states [9][10], decay of excited helium state [11], etc. Among these mechanisms one should mention frequent measurements which slow the evolution of a quantum system, hindering transitions to states different from the initial one.…”