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arXiv:1003.1645 · Physical Review E

Anomalous decay of a prepared state due to non-Ohmic coupling to the continuum

Itamar Sela, James Aisenberg, Tsampikos Kottos, Alexander Elgart, Doron Cohen

Published Mar 25, 2010• 2 citations
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Abstract

We study the decay of a prepared state E0 into a continuum {Ek} in the case of non-Ohmic models. This means that the coupling is Vk,|proportional|Ek-E0s-1 with s not equal 1. We find that irrespective of model details there is a universal generalized Wigner time t0 that characterizes the decay of the survival probability P0(t). The generic decay behavior which is implied by rate equation phenomenology is a slowing down stretched exponential, reflecting the gradual resolution of the band profile. But depending on nonuniversal features of the model a power-law decay might take over: it is only for an Ohmic coupling to the continuum that we get a robust exponential decay that is insensitive to the nature of the intracontinuum couplings. The analysis highlights the coexistence of perturbative and nonperturbative features in the dynamics. It turns out that there are special circumstances in which t0 is reflected in the spreading process and not only in the survival probability, contrary to the naive linear-response theory expectation. .

DOI
10.1103/physreve.81.036219

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We study the decay of a prepared state E0 into a continuum {Ek} in the case of non-Ohmic models. This means that the coupling is Vk,|proportional|Ek-E0s-1 with s not equal 1. We find that irrespective of model details there is a universal generalized Wigner time t0 that characterizes the decay of the survival probability P0(t). The generic decay behavior which is implied by rate equation phenomenology is a slowing down stretched exponential, reflecting the gradual resolution of the band profile. But depending on nonuniversal features of the model a power-law decay might take over: it is only for an Ohmic coupling to the continuum that we get a robust exponential decay that is insensitive to the nature of the intracontinuum couplings. The analysis highlights the coexistence of perturbative and nonperturbative features in the dynamics. It turns out that there are special circumstances in which t0 is reflected in the spreading process and not only in the survival probability, contrary to the naive linear-response theory expectation. .
We study the decay of a prepared state into non-flat continuum. We find that the survival probability P ( t ) might exhibit either stretched-exponential or power-law decay, depending on non-universal features of the model. Still there is a universal characteristic time t 0 that does not depend on the functional form. It is only for a flat continuum that we get a robust exponential decay that is insensitive to the nature of the intra-continuum couplings. The analysis highlights the co-existence of perturbative and non-perturbative features in the local density of states, and the nonlinear dependence of 1/ t 0 on the strength of the coupling.
Abstract It is known that quantum systems yield non‐exponential (power law) decay on long time scales, associated with continuum threshold effects contributing to the survival probability for a prepared initial state. For an open quantum system consisting of a discrete state coupled to continuum, we study the case in which a discrete bound state of the full Hamiltonian approaches the energy continuum as the system parameters are varied. We find in this case that at least two regions exist yielding qualitatively different power law decay behaviors; we term these the long time ‘near zone’ and long time ‘far zone’. In the near zone the survival probability falls off according to a t‐1 power law, and in the far zone i t falls off as t‐3. We show that the timescale TQ separating these two regions is inversely related to the gap between the discrete bound state energy and the continuum threshold. In the case that the bound state is absorbed into the continuum and vanishes, then the time scale TQ diverges and the survival probability follows the t‐1 power law even on asymptotic scales. Conversely, one could study the case of an anti‐bound state approaching the threshold before being ejected from the continuum to form a bound state. Again the t‐1 power law dominates precisely at the point of ejection.
We analyze the wave-packet dynamics scenario produced by Random Matrix Theory models with unconventional band-profile structures. Our motivation is to understand the energy spreading of quantum systems with complex dynamics. Examples of such systems include complex nuclei, atoms and molecules, quantum dots, and Bose-Einstein Condensates in optical traps, which, under the influence of an external perturbation, experience an energy redistribution of the initially prepared state. Such a perturbation could be due to an external electric or magnetic field, a change in the confining geometry, or a residual interaction, among other things. Of special interest in our analysis is the investigation of the time relaxation properties of a prepared state into a sea of other states (the continuum). We find that, for a large family of power spectra characterized by a non-flat profile, the survival probability P(t)P(t)P(t) might exhibit either exponential-like or power-law decay.
A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).
A metastable quantum state coupled to a continuum undergoes an exponential decay in the Weisskopf-Wigner (Markovian) approximation. However, quantum theory strictly predicts deviations from an exponential decay law both at small and long time scales. In multilevel systems, even in the Markovian approximation strong deviations from an exponential decay can be observed in the intermediate time scale owing to interference of overlapping resonances. Such interference effects are simply described by an effective non-Hermitian Hamiltonian and are known to explain, for example, the existence of dark states and fractional decay. Here we show that the wide variety of anomalous behaviors observed in multilevel quantum decay are rooted in the non-normal nature of the effective non-Hermitian Hamiltonian, revealing strong similarities between anomalous quantum mechanical decay and non-normal dynamics found in hydrodynamic flows. Major signatures of non-normal dynamics include delayed decay, accelerated decay, and exponential-power-law decay at an exceptional point. Such signatures are exemplified by suggesting simple tight-binding lattice realizations of multilevel quantum decay, which could be emulated in integrated photonic experiments. Finally, it is shown that for noninteracting particles with fermionic statistics the usual exponential decay law is restored, i.e., all anomalous decay effects arising from single-particle non-normal dynamics are washed out.
Results of theoretical studies of the quantum unstable systems caused that there are rather widespread belief that a universal feature of the quantum decay process is the presence of three time regimes of the decay process: the early time (initial) leading to the Quantum Zeno (or Anti Zeno) Effects, “exponential” (or “canonical”) described by the decay law of the exponential form, and late time characterized by the decay law having inverse-power law form. Based on the fundamental principles of the quantum theory we give the proof that there is no time interval in which the survival probability (decay law) could be a decreasing function of time of the purely exponential form but even at the “exponential” regime the decay curve is oscillatory modulated with a smaller or a large amplitude of oscillations depending on parameters of the model considered.