Long-time deviations from exponential decay for inverse-square potentials

Abstract

Quantal systems are predicted to show a change-over from exponential decay to power law decay at very long times. Although most theoretical studies predict integer power-law exponents, recent measurements by Rothe et al. of decay luminescence of organic molecules in solution {Phys. Rev. Lett. 96 (2006) 163601} found non-integer exponents in most cases. We propose a physical mechanism, within the realm of scattering from potentials with long tails, which produces a continuous range of power law exponents. In the tractable case of the repulsive inverse square potential, we demonstrate a simple relation between the strength of the long range tail and the power law exponent. This system is amenable to experimental scrutiny.

Enhanced observability of quantum postexponential decay using distant detectors

“…Aside from the non-trivial challenge of understanding and extending these experimental results, there are many reasons for studying post-exponential decay: Winter, for example, argued that hidden-variable theories could produce observable effects in samples that have decayed for many life-times [13]; more recently Krauss and Dent have pointed out that late-time decay may have important cosmological implications [14]; decay at long times is quite * Electronic address: jg.muga@ehu.es † Electronic address: martorell@ecm.ub.es sensitive to delicate measurement and/or environmental effects, so it is a testing ground for theories of these processes; at a fundamental level, the form of long time deviations from exponential decay might distinguish between standard, Hermitian quantum mechanics [15], and modifications with a built-in microscopic arrow of time [16,17]; from a practical perspective, Norman pointed out that post-exponential decay could set a limit to the validity of radioactive dating methods [9]; we have also argued that the deviation could be used to characterize certain cold atom traps [18]. Indeed, due to technological advances in lasers, semiconductors, nanoscience, and cold atoms, microscopic interactions are now relatively easy to manipulate: this makes decay parameters controllable, and post-exponential decay more accessible to experimental scrutiny and/or applications.…”

Resonant spectra of quadrupolar anions

“…Moreover, rotational states in dipolar anions were expected to be strongly affected by the shallowness of the molecular potential and the nonadiabatic coupling of the electronic and molecular rotational motions [27,48,[55][56][57][58]. The strong coupling of the attached electron to the continuum [59][60][61][62][63][64][65] renders the picture even more complex, with the existence of low-energy sharp resonances [45,[66][67][68][69][70][71] in various systems, and the modification of the Wigner's law [72][73][74] for the dipolar field [54,[75][76][77], first observed in hydrogen atoms [78,79] and then extended to different power-law potentials [75,76].…”

Scale-free relaxation of a wave packet in a quantum well with power-law tails

“…We will, however, see that, given its asymptotic scaling, the precise form of the confining well is not crucial for the predicted power-law decay of the survival probability. The potential of equation (1) has the same asymptotic behavior as in [22,23]; however, differently from the one considered in [23] it extends over the entire real axis, implying the existence of an equilibrium or ground state.…”

“…For our purpose the distribution of participating energies in the initial state should be continuously connected to zero and be sufficiently broad, as we will discuss below in more detail. In contrast to [23], we are explicitly interested in the survival probability P(t) that the particle remains confined within the well, i.e.…”

“…The proper choice of the confining potential allows one to induce algebraic rather than exponential decay, for generic, confined initial states. As described in [22,23], the crucial ingredient will be an asymptotically scaleinvariant potential decaying to zero as ∼x −2 , which occurs naturally in systems with dipolar interactions [24] or anomalous molecular binding potentials [25], and is often associated with quite counterintuitive effects (see, e.g., [22,[26][27][28][29]). We thus provide in section 2 a particularly transparent example of a scale-free relaxation process, where the equilibrium state is reached only at asymptotically long times [3,30].…”