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arXiv:quant-ph/0105138 · Physical Review A

Real measurements and the quantum Zeno effect

Julius Ruseckas, B. Kaulakys

Published May 8, 2001• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.63.062103

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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.
Summary form only given. The quantum Zeno effect (QZE) is the striking possibility of inhibiting the decay of an unstable quantum state by sufficiently frequent measurements. This effect is currently believed to be universal, even though it has only been observed for oscillations between discrete states. Recently we showed that both the QZE and the inverse (anti-) Zeno effect (AZE) can be observed on spontaneous decay in cavities. We develop a general theory of evolution of quantum systems in the presence of frequent measurements. We consider both the ideal case of instantaneous projections of the wave function to the initial state and some possible measurement schemes. The considerations imply the impossibility of the continuous-observation limit necessary to stop quantum evolution (the quantum Zeno paradox). Indeed, this limit leads to an unbounded increase of the energy fluctuations of the system, which would cause the system to disintegrate. The present theory describes different measurement effects, including the QZE and AZE, and allows us to obtain the general criteria of their validity. In particular, we show that the AZE, rather than the QZE, is achievable for spontaneous decay of atoms in free space.
Itano and co-workers [Wayne M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295 (1990)] have recently reported the experimental verification of the quantum Zeno effect, which is the inhibition of a quantum transition by frequent measurements. In this article, we offer an alternative interpretation of the quantum Zeno effect. We show that an analysis of the dynamics of the full three-level system gives the same result. There is no need to assume explicitly that the wave function has collapsed, nor even to assume that an ideal measurement has been made. In addition, we differentiate between what has been referred to as the quantum Zeno effect and what has been termed the quantum Zeno paradox. The former is the inhibition of induced transitions, and the latter is the, as yet experimentally unobserved, inhibition of spontaneous decay. Our interpretation, which emphasizes the ``measurement''-induced interruption of atomic-state coherences as the cause of inhibited quantum transitions, suggests a resolution to the quantum Zeno paradox. The theoretical limit of continuous observation is discussed.
We study the quantum Zeno effect and the anti-Zeno effect in the case of ``indirect'' measurements, where a measuring apparatus does not act directly on an unstable system, for a realistic model with finite errors in the measurement. A general and simple formula for the decay rate of the unstable system under measurement is derived. In the case of a Lorentzian form factor, we calculate the full time evolutions of the decay rate, the response of the measuring apparatus, and the probability of errors in the measurement. It is shown that not only the response time but also the detection efficiency plays a crucial role. We present the prescription for observing the quantum Zeno and anti-Zeno effects, as well as the prescriptions for avoiding or calibrating these effects in general experiments.
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.
A simple model of indirect pre-measurements on an unstable quantum state is presented in this paper. The model is completely solvable and the solutions are used to compare the time evolution of the unstable state with and without the influence of the pre-measurement. We find that by choosing the details of the process of pre-measurement appropriately, it is possible to observe both suppression and enhancement of the rate of decay of the unstable state. When the pre-measurements are assumed to lead on to actual measurements, we see that the quantum Zeno effect, and in some instances the 'anti-Zeno' effect, can be produced by repeating the measurement many times in succession. The Zeno effect can appear in our model either as a real consequence of repeated measurements or sometimes merely as an artefact of the manner in which the observations on the system are performed. The anti-Zeno effect appears almost exclusively as an artefact of the details of the measurement. Numerical investigations are included to delineate the regimes in which the quantum Zeno effect and possibly the anti-Zeno effect can occur.