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QUANTUM ZENO EFFECT FOR N-LEVEL FRIEDRICHS MODEL

Ioannis E. Antoniou, Evgueni Karpov, G. P. Pronko, E. A. Yarevsky

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

We study the short-time behaviour of the survival probability in the framework of the N -level Friedrichs model. We show that depending on initial conditions the decay can be considerably slowed down or even stopped. By choosing proper parameters of the system, the Zeno time can also be considerably extended.

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arXiv:2508.13194•Symmetry•Aug 15, 2025•2 citations
We study the short-time and medium-time behavior of the survival probability of decaying states in the framework of the NNN-level Friedrichs model. The degenerated and nearly degenerated systems are discussed in detail. We show that in these systems decay can be considerably slowed down or even stopped by appropriate choice of initial conditions. We analyze the behaviour of the multilevel system within the so-called Zeno era. We examine and compare two different definitions of the Zeno time. We demonstrate that the Zeno era can be considerably enlarged by proper choice of the system parameters.
Quantum mechanics enables the calculation of the time evolution of a dynamical system provided the Hamiltonian is defined and the initial state is specified. Due to the time reversal invariance, the time evolution of a given initial state at t=0 does not have a time-arrow. These problems have been extensively studied by E. C. G. Sudarshan and collaborators, including myself. Here the survival probability, which is the square of the survival amplitude, is an even function of t. Using the survival amplitude as a common framework, we discuss Zeno effects in several processes: the decay of an unstable particle, pion productions in proton-nucleus collisions and the Aharonov-Vardi effect which is another manifestation of the Zeno effect.
We study the time evolution of decaying particles in the framework of certain renormalizable models of relativistic quantum field theory. Significant differences between the latter and the non-relativistic quantum mechanical case are found —in particular, regarding the so-called quantum Zeno effect. In some cases, the finite time behaviour is well defined after having performed a conventional renormalization. However, in other situations, the latter does not suffice to obtain finite survival probabilities. In all cases, the crucial consistency with Fermi’s golden rule holds. The analysis of the finite time evolution in relativistic quantum field theory (RQFT) and, in particular, the behaviour at short times of the survival probability of unstable particles in RQFT is a relevant matter as far as the foundations of quantum mechanics are concerned and has some peculiarities which make it to be worth studying. But before going to RQFT, we will briefly recapitulate the main features of the survival amplitude (SA) at
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.