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Physical Review A

Short- and long-time decay laws and the energy distribution of a decaying state

K. Unnikrishnan

Published Jul 1, 1999• 9 citations
Publisher

Abstract

The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schr\"odinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as ${t}^{{\ensuremath{-}}^{n}}$ if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.

DOI
10.1103/physreva.60.41

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The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schr"odinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as t\ensuremath−n{t}^{{\ensuremath{-}}^{n}}t\ensuremath−n if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.
The short-time behavior of quantum decay of an unstable state initially located within an interaction region of finite range is investigated using a resonant expansion of the survival amplitude. It is shown that in general the short-time behavior of the survival probability S(t)S(t)S(t) has a dependence on the initial state and may behave as either $S(t)=1\ensuremath{-}O({t}^{3/2})$ or $1\ensuremath{-}O({t}^{2})$. These cases are illustrated by solvable models. The experiment reported by Wilkinson et al. [Nature (London) 387, 575 (1997)] does not distinguish between the above short-time behaviors.
The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.
It has been shown for various systems that the decay rate of an unstable quantum system deviates from exponential behavior in short- and long-time regimes associated with memory effects. In particular, it is widely believed that in the short-time regime, the decay is quadratic, inducing the quantum Zeno effect, in which the decay is suppressed by rapidly repeated measurements. In our study, we find that when the environment of the unstable system has an energy spectrum with a lower bound but without an upper one, the decay rate in both regimes is scaled in terms of the spatial dimension and the exponent of the energy dispersion of the environment. Surprisingly, we find that in the short-time regime the decay exhibits fractional scaling, which leads to a quantum Zeno effect with a different scaling of the Zeno time.
We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude a(t)a(t)a(t) of the state considered is found. Our attention is focused on the late time properties of a(t)a(t)a(t) and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as ttt goes to the infinity more slowly than any exponential function of ttt. This effect can be described using a time-dependent decay rate γ(t)γ(t)γ(t) and then the Khalfin result means that this γ(t)γ(t)γ(t) is not a constant but at late times it tends to zero as ttt goes to the infinity. It appears that the energy E(t)E(t)E(t) of the unstable state behaves similarly: It tends to the minimal energy EminE_{min}Emin​ of the system as t→∞t \to \inftyt→∞. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of E(t)E(t)E(t) and γ(t)γ(t)γ(t). We discuss also possible implications of such a late time asymptotic properties of E(t)E(t)E(t) and γ(t)γ(t)γ(t) and cases where these properties may manifest themselves.