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arXiv:math-ph/0307044

Mathematics of the Quantum Zeno Effect

Schmidt, Andreas U.

Published Jul 21, 2003• 7 citations
arXivHTMLPDF

Abstract

We present an overview of the mathematics underlying the quantum Zeno effect. Classical, functional analytic results are put into perspective and compared with more recent ones. This yields some new insights into mathematical preconditions entailing the Zeno paradox, in particular a simplified proof of Misra's and Sudarshan's theorem. We empahsise the complex-analytic structures associated to the issue of existence of the Zeno dynamics. On grounds of the assembled material, we reason about possible future mathematical developments pertaining to the Zeno paradox and its counterpart, the anti-Zeno paradox, both of which seem to be close to complete characterisations.

DOI
10.48550/arxiv.math-ph/0307044

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We present an overview of the mathematics underlying the quantum Zeno effect. Classical, functional analytic results are put into perspective and compared with more recent ones. This yields some new insights into mathematical preconditions entailing the Zeno paradox, in particular a simplified proof of Misra's and Sudarshan's theorem. We empahsise the complex-analytic structures associated to the issue of existence of the Zeno dynamics. On grounds of the assembled material, we reason about possible future mathematical developments pertaining to the Zeno paradox and its counterpart, the anti-Zeno paradox, both of which seem to be close to complete characterisations.
In the 5th century B.C.,the philosopher and logician Zeno of Elea posed several paradoxes which remained unresolved for over two thousand five hundred years. The 20th20^{th}20th century saw some resolutions to Zeno's mind boggling problems. This long journey saw many significant milestones in the form of discoveries like the tools of converging series and theories on infinite sets in mathematics. In recent times, the Zeno effect made an intriguing appearance in a rather unlikely place - a situation involving the time evolution of a quantum system, which is subject to "observations" over a period of time. Leonid Khalfin working in the former USSR in the 1960s and ECG Sudarshan and B. Misra at the University of Texas, Austin, first drew attention to this problem. In 1977, ECG Sudarshan and B. Misra published a paper on the quantum Zeno effect, called "The Zeno's paradox in quantum theory". Their fascinating result revealed the bizarre workings of the quantum world. Misra and Sudarshan's 1977 paper activated over two decades of theoretical and experimental explorations into the subject and still continues to evoke a lot of interest. In the following, the quantum Zeno effect is described and a brief outline of some of the work following Misra and Sudarshan's paper is given. The quantum Zeno effect is yet another example of the myriad unimaginable possibilities that lie waiting in the magical world of the quantum.
As of October 2006, there were approximately 535 citations to the seminal 1977 paper of Misra and Sudarshan that pointed out the quantum Zeno paradox (more often called the quantum Zeno effect). In simple terms, the quantum Zeno effect refers to a slowing down of the evolution of a quantum state in the limit that the state is observed continuously. There has been much disagreement as to how the quantum Zeno effect should be defined and as to whether it is really a paradox, requiring new physics, or merely a consequence of "ordinary" quantum mechanics. The experiment of Itano, Heinzen, Bollinger, and Wineland, published in 1990, has been cited around 347 times and seems to be the one most often called a demonstration of the quantum Zeno effect. Given that there is disagreement as to what the quantum Zeno effect is , there naturally is disagreement as to whether that experiment demonstrated the quantum Zeno effect. Some differing perspectives regarding the quantum Zeno effect and what would constitute an experimental demonstration are discussed.
The term 'quantum Zeno paradox' or 'quantum Zeno effect' refers to the slowing down of the evolution of a quantum system as it is observed more and more frequently.In 1977, Misra and Sudarshan gave a theoretical demonstration of its existence.There has been disagreement regarding the definition of the effect and whether it is really a paradox, requiring new physics, or a consequence of ordinary quantum mechanics.The experiment of Itano, Heinzen, Bollinger and Wineland, published in 1990, is the one most often called a demonstration of the quantum Zeno effect.Some different interpretations are discussed.
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.
Introduction Quantum theory predicts the striking and paradoxical result that when a system is continuously watched, it does not evolve! Although this effect was noticed much earlier by a number of people [146], [147], [148], [149], [150], it was first formally stated by Misra and Sudarshan [151] and given the appellation ‘Zeno's paradox’ because it evokes the famous paradox of Zeno denying the possibility of motion to a flying arrow. It is as startling as a pot of water on a heater that refuses to boil when continuously watched. This is why it is also called the ‘watched pot effect’. The effect is the result of repeated, frequent measurements on the system, each measurement projecting the system back to its initial state. In other words, the wave function of the system must repeatedly collapse. It is also necessary that the time interval between successive measurements must be much shorter than the critical time of coherent evolution of the system, called the Zeno time [152]. For decays this Zeno time is the time of coherent evolution before the irreversible exponential decay sets in, and is governed by the reciprocal of the range of energies accessible to the decay products. For most decays this is extremely short and hard to detect [153]. In the case of non-exponential time evolution, the relevant Zeno time can be much longer.