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European Journal of Physics

The quantum Zeno effect in double well tunnelling

L Lerner

Published Jan 18, 2018• 2 citations
Publisher

Abstract

Abstract Measurement lies at the heart of quantum theory, and introductory textbooks in quantum mechanics cover the measurement problem in topics such as the Schrödinger’s cat thought experiment, the EPR problem, and the quantum Zeno effect (QZE). In this article we present a new treatment of the QZE suitable for undergraduate students, for the case of a particle tunnelling between two wells while being observed in one of the wells. The analysis shows that as the observation rate increases, the tunnelling rate tends towards zero, in accordance with Zeno’s maxim ‘a watched pot never boils’. The method relies on decoherence theory, which replaces aspects of quantum collapse by the Schrödinger evolution of an open system, and its recently simplified treatment for undergraduates. Our presentation uses concepts familiar to undergraduate students, so that calculations involving many-body theory and the formal properties of the density matrix are avoided.

DOI
10.1088/1361-6404/aaa8a5

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Abstract Measurement lies at the heart of quantum theory, and introductory textbooks in quantum mechanics cover the measurement problem in topics such as the Schrödinger’s cat thought experiment, the EPR problem, and the quantum Zeno effect (QZE). In this article we present a new treatment of the QZE suitable for undergraduate students, for the case of a particle tunnelling between two wells while being observed in one of the wells. The analysis shows that as the observation rate increases, the tunnelling rate tends towards zero, in accordance with Zeno’s maxim ‘a watched pot never boils’. The method relies on decoherence theory, which replaces aspects of quantum collapse by the Schrödinger evolution of an open system, and its recently simplified treatment for undergraduates. Our presentation uses concepts familiar to undergraduate students, so that calculations involving many-body theory and the formal properties of the density matrix are avoided.
Observe an unstable quantum system frequently enough and its decay slows, and in the limit of continuous observation stops altogether. This is measurement-induced state freezing — the quantum Zeno effect, the physicist's "a watched pot never boils." It was named by Misra and Sudarshan (1977) and measured by Itano, Heinzen, Bollinger and Wineland in a ion trap (1990). It is standard, established quantum mechanics and is not in dispute. This paper does three things. (i) It restates, from first principles and with a numerical runner, why the effect exists at all: it is a statement about the exponent of the short-time survival law, and about nothing else. Frequent projective measurement can suppress a loss that grows quadratically in time (a coherent two-state amplitude, survival ), and cannot suppress a loss that grows linearly in time (a memoryless classical rate, survival ). (ii) It reports the two cases side by side, matched to agree when unwatched and then watched, together with a fitted short-time exponent — 1.9999 for the coherent case, 0.9967 for the classical case — and a "look but do not reset" control. (iii) In a clearly labelled interpretive section it examines the Temporal Dynamics Framework's account of the effect and shows that this account is a fork: the framework's fundamental object [the flip — in standard vocabulary a two-state transition of the underlying oscillation, which may be read either as a coherent amplitude or as a classical rate] must be committed to one reading, and the two readings give opposite answers. Only the coherent reading reproduces Zeno. This record holds one paper on its own and is refreshed every couple of months. The complete deposit, 10.5281/zenodo.21788037, holds every paper with its code and data and is updated first; for the newest version of this paper, go there.
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.
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
We present the physics of the quantum Zeno effect, whose gist is often expressed by invoking the adage "a watched pot never boils". We review aspects of the theoretical and experimental work done on the effect since its inception in 1977, and mention some applications. We dedicate the article - with our very best wishes - to Rudolf Zahradnik at the occasion of his great jubilee. Perhaps Rudolf's lasting youthfulness and freshness are due to that he himself had been frequently observed throughout his life: until the political turn-around in 1989 by those who wished, by their surveillance, to prevent Rudolf from spoiling the youth by his personal culture and his passion for science and things beautiful and useful in general. This attempt had failed. Out of gratitude, the youth has infected Rudolf with its youthfulness. Chronically. Since 1989, Rudolf has been closely watched by the public at large. For the same traits of his as before, but with the opposite goal and for the benefit of all generations. We relish keeping him in sight ...
We propose an experiment for the observation of the quantum Zeno effect (QZE) in a bipartite system. The setup involves two microwave cavities and a ``tunneling'' photon, which is observed by the passage of Rydberg atoms. Our proposal allows for the consideration of two types of measurements, namely, sequential observations of the atomic state and its inclusive measurement. In the present system the two processes are shown to lead to the same result in the ideal case. We consider realistic atom-field interaction times, cavity dissipation, and limited detection efficiency. Analytical expressions for the ``tunneling'' probability are obtained exhibiting a competition between the environment induced exponential decay and the characteristic t2{t}^{2}t2 (for short times) dependence of the QZE. We show that for sufficiently small dissipation constants the effect can be observed with current experimental facilities.