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arXiv:0903.4156 · Physical Review A

Enhanced observability of quantum postexponential decay using distant detectors

E. Torrontegui, Juan Gonzalo Muga, Joan Martorell, Donald W. L. Sprung

Published Jul 10, 2009• 22 citations
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Abstract

We study the elusive transition from exponential to postexponential (algebraic) decay of the probability density of a quantum particle emitted by an exponentially decaying source in one dimension. The main finding is that the probability density at the transition time, and thus its observability, increases with the distance of the detector from the source up to a critical distance beyond which exponential decay is no longer observed. Solvable models provide explicit expressions for the dependence of the transition on resonance and observational parameters, facilitating the choice of optimal conditions.

DOI
10.1103/physreva.80.012703

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We study the elusive transition from exponential to postexponential (algebraic) decay of the probability density of a quantum particle emitted by an exponentially decaying source in one dimension. The main finding is that the probability density at the transition time, and thus its observability, increases with the distance of the detector from the source up to a critical distance beyond which exponential decay is no longer observed. Solvable models provide explicit expressions for the dependence of the transition on resonance and observational parameters, facilitating the choice of optimal conditions.
Postexponential decay of the probability density of a quantum particle leaving a trap can be reproduced accurately, except for interference oscillations at the transition to the postexponential regime, by means of an ensemble of classical particles emitted with constant probability per unit time and the same half-life as the quantum system. The energy distribution of the ensemble is chosen to be identical to the quantum distribution, and the classical point source is located at the scattering length of the corresponding quantum system. A one-dimensional example is provided to illustrate the general argument.
We study the decay of unstable states by formulating quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time-of-arrival determined by Anastopoulos and Savvidou [J. Math. Phys. 47, 122106 (2006)]. This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that the exponential decay only arises if three specific mathematical conditions are met. Their physical content is the following: (i) the decay time is much larger than any microscopic timescale, so that the fine details of the initial state can be ignored, (ii) there is no quantum coherence between the different “attempts” of the particle to traverse the barrier, and (iii) the transmission probability varies little within the momentum spread of the initial state. We also determine the long time limits of the decay probability and we identify regimes, in which the decays have no exponential phase.
We derive an analytical expression for the propagation of a quantum particle at asymptotic long distances and times from the potential where the particle was initially confined. We obtain that the particle is described by an evolving resonance in time domain that possesses a peaked shape characterized by the resonance parameters of the dominant resonance coefficient of the decaying wave function. The above situation corresponds to an unexplored postexponential regime that represents the ultimate fate of a decaying particle.
Quantal systems are predicted to show a changeover from exponential to a slower decay rate at very long times. Asymptotically most models predict power-law decay with integer exponents. However, the postexponential decay of a trapped particle from a potential can occur with a continuous range of power-law exponents. We show that this happens when the outer part of the potential is repulsive and decreases with the inverse square of the distance. We demonstrate a simple relation between the strength of the long-range tail and the power-law exponent. We also give explicit forms for the postexponential decay laws when the outer part of the potential is weakly attractive. These systems are amenable to experimental scrutiny.
Abstract Methods based on the use of Green’s functions or Jost functions and the Fock–Krylov method are apparently very different approaches to understand the time evolution of unstable states. We show that the two former methods are equivalent up to some constants and as an outcome find an analytic expression for the energy density of states in the Fock–Krylov amplitude in terms of the coefficients introduced in the Green’s functions and the Jost functions methods. This model-independent density is further used to obtain an analytical expression for the survival amplitude and study its behaviour at large times. Using these expressions, we investigate the origin of the oscillatory behaviour of the decay law in the region of the transition from the exponential to the non-exponential at large times. With the objective of understanding the failure of nuclear and particle physics experiments in observing the non-exponential decay law predicted by quantum mechanics for large times, we derive analytical formulae for the critical transition time, , from the exponential to the inverse power law behaviour at large times. Evaluating for some particle resonances and narrow nuclear states which have been tested experimentally to verify the exponential decay law, we conclude that the large time power law in particle and nuclear decay is hard to find experimentally.