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Journal of Physical Studies

Nonuniform quantum Zeno effect

V. M. Tkachuk

Published Jan 1, 2011• 1 citation
Publisher

Abstract

We study the quantum Zeno effect in the case when the time intervals between two consecutive measurements are different.It can be called a nonuniform quantum Zeno effect.Explicit examples of the nonuniform quantum Zeno effect are presented.We also show that maximal quantum Zeno effect in the case of a large number of measurements can be achieved when the time intervals between consecutive measurements are equal.

DOI
10.30970/jps.15.1001

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We study the quantum Zeno effect in the case when the time intervals between two consecutive measurements are different.It can be called a nonuniform quantum Zeno effect.Explicit examples of the nonuniform quantum Zeno effect are presented.We also show that maximal quantum Zeno effect in the case of a large number of measurements can be achieved when the time intervals between consecutive measurements are equal.
If one continuously measures a decaying system, the system will appear to never decay that was called quantum Zeno effect. The continuous measurement is defined by a sequence of measurements whose time interval t between measurements approaches zero. Later many works chose the time interval t as finite (and greater than the Zeno time) which corresponds to making equal spaced measurements over a discrete time interval. With the discrete variable formulism one can derive the so-called Anti-Zeno effect. Our study is trying to contrast the results between continuous time interval measurement versus discrete time interval measurement. We demonstrate that we can obtain so-called “Zeno” and “Anti-Zeno” in a classical system if we apply the definition of non-ideal measurement.
Abstract. The evolution of a quantum system subjected to infinitely many measurements in a finite time interval is confined in a proper subspace of the Hilbert space. This phenomenon is called “quantum Zeno effect”: a particle under intensive observation does not evolve. This effect is at variance with the classical evolution, which obviously is not affected by any observations. By a semiclassical analysis we will show that the quantum Zeno effect vanishes at all orders, when the Planck constant tends to zero, and thus it is a purely quantum phenomenon without classical analog, at the same level of tunneling. PACS numbers: 03.65.Sq, 03.65.Xp ar
In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such non-selective measurements. Importantly, we show that there is a difference between using non-selective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems, and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective measurements can differ very significantly as compared to the repeated selective measurement scenario.
In this work we consider a new example of the recently introduced quantum Hamlet effect. We consider an especial, abstract, unstable quantum system whose evolution during a small time interval is interrupted by frequent measurements. Here three different final situations exist. First one corresponds to quantum Zeno effect, second one - to quantum anti-Zeno effect and third one - to so-called quantum Hamlet effect. By quantum Zeno effect final non-decay probability is function of number of the decay measurements variable and dynamical degree parameter equivalent to two. When measurements number tends toward infinity non-decay probability has the one limit, or, it tends analytically toward one and system stands non-decayed. By quantum anti-Zeno effect final non-decay probability is function of number of the decay measurements variable and dynamical degree parameter equivalent to one. When measurements number tends toward infinity non-decay probability has the zero limit, or, it tends analytically toward zero and system becomes decayed. By quantum Hamlet effect, final non-decay probability is function of two variable, number of the decay measurements and dynamical degree. When measurements number tends toward infinity and dynamical degree toward one, final non-decay probability depends not only of final value of given variables, but, also, on the ways on which given variables tends toward their final values. It means that final no-decay probability has not (analytical) limit, or that there is no {\it analytical} prediction on the final no-decay probability. To be decayed or no-decayed that is analytically unsolvable question for given quantum system.