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arXiv:1702.02250 · Physical Review A

Nonexponential tunneling decay of a single ultracold atom

Gastón García‐Calderón, Roberto Romo

Published Feb 25, 2016• 17 citations
arXivHTMLPDF

Abstract

By using an exact analytical approach to the time evolution of decay we investigate the tunneling decay of ultracold single atoms to discuss the conditions for deviations of the exponential decay law. We find that $R$, given by the ratio of the energy of the decaying fragment ${\mathcal{E}}_{r}$ to its corresponding width ${\mathrm{\ensuremath{\Gamma}}}_{r}$, is the relevant quantity in this study. When $R$ is less than $0.3$, the decay of the atom goes to a good approximation for the first few lifetimes as $exp(\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{r}t/2\ensuremath{\hbar}){t}^{\ensuremath{-}3/2}$. We also find that for values of $R\ensuremath{\sim}1$, the nonexponential behavior occurs in a postexponential regime that goes as ${t}^{\ensuremath{-}3}$ after around a dozen lifetimes. The above conditions depend on suitably designed potential parameters and suggest that for values $R\ensuremath{\lesssim}1$, the experimental verification of nonexponential decay might be possible.

DOI
10.1103/physreva.93.022118

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By using an exact analytical approach to the time evolution of decay we investigate the tunneling decay of ultracold single atoms to discuss the conditions for deviations of the exponential decay law. We find that RRR, given by the ratio of the energy of the decaying fragment Er{\mathcal{E}}_{r}Er​ to its corresponding width \ensuremathΓr{\mathrm{\ensuremath{\Gamma}}}_{r}\ensuremathΓr​, is the relevant quantity in this study. When RRR is less than 0.30.30.3, the decay of the atom goes to a good approximation for the first few lifetimes as exp(\ensuremath−\ensuremathΓrt/2\ensuremathℏ)t\ensuremath−3/2exp(\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{r}t/2\ensuremath{\hbar}){t}^{\ensuremath{-}3/2}exp(\ensuremath−\ensuremathΓr​t/2\ensuremathℏ)t\ensuremath−3/2. We also find that for values of R\ensuremath∼1R\ensuremath{\sim}1R\ensuremath∼1, the nonexponential behavior occurs in a postexponential regime that goes as t\ensuremath−3{t}^{\ensuremath{-}3}t\ensuremath−3 after around a dozen lifetimes. The above conditions depend on suitably designed potential parameters and suggest that for values R\ensuremath≲1R\ensuremath{\lesssim}1R\ensuremath≲1, the experimental verification of nonexponential decay might be possible.
Summary form only given.We report the observation of short-time deviation from exponential decay in a quantum tunneling experiment. Our system consists of ultracold sodium atoms that are trapped in an accelerating periodic optical potential created by a standing wave of light. Atoms can escape the wells via quantum tunneling, and the number that remain is measured as a function of interaction time for a fixed value of the well depth and acceleration. To study tunneling in this system, sodium atoms are first trapped and cooled in a magneto-optical trap. The trapping beams as well as the magnetic field gradient are then turned off, and an accelerating far-detuned standing wave of light is turned on. A three-step acceleration is implemented to measure the loss due to tunneling. Once the final velocity is reached, the potential is turned off and the atomic fluorescence is imaged after a free drift time. We have measured the survival probability and found that the initial survival probability is flat and then crosses over to an exponential decay. The intermediate stage is characterized by a time-dependent loss rate. To analyze the short-time tunneling probability and the approach to exponential decay, we use a simple model of the band structure with a single (trapped) band separated by a bandgap from a free particle state. We find an analytic expression for the survival probability that is quadratic for the first few microseconds of evolution, with a crossover to exponential decay at long times. We show that this crossover time is related to the tunneling time. We also identify a new parameter regime where coherent oscillations of the survival probability are seen.
The exponential and nonexponential regimes of the buildup process in resonant tunneling structures are analyzed by considering an analytic solution of the time-dependent Schr\"odinger equation. It is found that the buildup exhibits a purely exponential behavior in a finite time interval followed by a clear transition to a nonexponential regime. The buildup of the probability amplitude in the nonexponential regime follows a ${t}^{\ensuremath{-}3/2}$ time dependence, in the same fashion as the survival amplitude in quantum decay. For incidence energies at higher isolated resonances, $E={\ensuremath{\varepsilon}}_{n}$ with n>1n>1n>1, it is found that the exponential regime is split into two exponential subregimes: the first one is governed by the width ${\ensuremath{\Gamma}}_{n}$ of the resonance chosen for the incidence energy, and the second one is dominated by the width ${\ensuremath{\Gamma}}_{1}$ of the lowest resonance. The transition occurs directly from nnn to 1 without jumps to other intermediate resonant states. This dynamics is discussed in comparison with the (opposite) and well-studied process of quantum decay, with which we find that there are striking similarities in both the exponential and nonexponential regimes. We also analyze the buildup in systems with resonance doublets, where the interference effects produced by the interacting resonances play an important role. In the latter case we find that the buildup of the probability amplitude exhibits a complex oscillatory behavior that can be characterized by a mix of oscillating contributions with well-defined Rabi type frequencies of the form $|E\ensuremath{-}{\ensuremath{\varepsilon}}_{n}|/\ensuremath{\hbar}$, where ${\ensuremath{\varepsilon}}_{n}$ are the nearest resonances around the incidence energy EEE.
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.
We examine an analytical expression for the survival probability for the time evolution of quantum decay to discuss a regime where quantum decay is nonexponential at all times. We find that the interference between the exponential and nonexponential terms of the survival amplitude modifies the usual exponential decay regime in systems where the ratio of the resonance energy to the decay width is less than 0.3. We suggest that such a regime could be observed in semiconductor double-barrier resonant quantum structures with appropriate parameters.
While exponential decay is ubiquitous in nature, deviations at both short and long times are dictated by quantum mechanics. Nonexponential decay is known to arise due to the possibility of reconstructing the initial state from the decaying products. We discuss the quantum decay dynamics by tunneling of a many-particle system, characterizing the long-time nonexponential behavior of the nonescape and survival probabilities. The effects of contact interactions and quantum statistics are described. It is found that, whereas for noninteracting bosons the long-time decay follows a power law with an exponent linear in the number of particles NNN, the exponent becomes quadratic in NNN in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.
The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the nonexponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through specific quantum mechanical examples which can be accurately solved, we study the general properties of a two-channel decay that apply for various unstable quantum states (including unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit), but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential-decay law.