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arXiv:quant-ph/0012130 · Physical Review A

Quantum Zeno and anti-Zeno effects in the Friedrichs model

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

Published May 16, 2001• 35 citations
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Abstract

We analyze the short-time behavior of the survival probability in the frame of the Friedrichs model for different form factors. We have shown that this probability is not necessarily analytic at the time origin. The time when the quantum-Zeno effect could be observed is found to be much smaller than usually estimated. We have also studied the anti-Zeno period and have estimated its duration.

DOI
10.1103/physreva.63.062110

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We analyze the short-time behavior of the survival probability in the frame of the Friedrichs model for different form factors. We have shown that this probability is not necessarily analytic at the time origin. The time when the quantum-Zeno effect could be observed is found to be much smaller than usually estimated. We have also studied the anti-Zeno period and have estimated its duration.
If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, t∝Nα. The limit survival probability exhibits a sudden jump from 1 to 0 at α=1/2, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for α≥1, the limit probability becomes sensitive to the spectral properties of the initial state and to the arithmetic properties of the measurement periods.
In 1977, Mishra and Sudarshan [J. Math. Phys. 18, 756 (1977)] showed that an unstable particle would never be found decayed while it was continuously observed. They called this effect the quantum Zeno effect (or paradox). Later it was realized that the frequent measurements could also accelerate the decay (quantum anti-Zeno effect). In this paper, we investigate the quantum Zeno effect using the definite model of the measurement. We take into account the finite duration and the finite accuracy of the measurement. A general equation for the jump probability during the measurement is derived. We find that the measurements can cause inhibition (quantum Zeno effect) or acceleration (quantum anti-Zeno effect) of the evolution, depending on the strength of the interaction with the measuring device and on the properties of the system. However, the evolution cannot be fully stopped.
The quantum Zeno effect consists in the hindrance of the evolution of a quantum system that is very frequently monitored and found to be in its initial state at every single measurement. On the basis of the correct formula for the survival probability, i.e. the probability of finding the system in its initial state at every single measurement, we critically analyze a recent proposal and experimental test, that make use of an oscillating system. PACS: 03.65.Bz, 42.50.-p 1 The seminal formulation of the quantum Zeno effect, due to Misra and Sudarshan [1], deals with the probability of observing an unstable system in its initial state throughout a time interval ∆ = [0, t]. The purpose of this note is to point out that the quantum Zeno effect has not been experimentally observed, yet, in its original formulation. Indeed, we shall argue that the interesting proposal by Cook [2], that makes use of a two-level system undergoing Rabi oscillations, as well as the beautiful experiment performed by Itano et al. [3], investigate the probability of finding the
We study the time evolution of decaying particles in renormalizable models of Relativistic Quantum Field Theory. Significant differences between the latter and Non Relativistic Quantum Mechanics are found —in particular, the Zeno effect seems to be absent in such RQFT models. Conventional renormalization yields finite time behaviour in some cases but fails to produce finite survival probabilities in others. 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 short times in non relativistic quantum mechanics (QM henceforth); see, for instance, [1] for a comprehensive review.