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On a Simple Derivation of the Effect of Repeated Measurements on Quantum Unstable Systems by Using the Regularized Incomplete beta-Function

Elio Conte

Published Jan 1, 2012• 1 citation
OpenAlex

Abstract

a simple derivation of the effect induced from repeated measurements on quantum unstable systems is obtained by using the regularized incomplete beta - function.

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On a Simple Derivation of the Effect of Repeated Measurements on Quantum Unstable Systems by Using the Regularized Incomplete

Elio Conte

a simple derivation of the effect induced from repeated measurements on quantum unstable systems is obtained by using the regularized incomplete  - function .

•Jan 1, 2012•0 citations

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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.

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The quantum-mechanical time evolution of an isolated \ensuremath{\beta}-unstable nuclear state in the context of certain models predicts that the square of the amplitude of the initial undecayed state becomes an approximately exponential function of time on a timescale on the order of ${10}^{\ensuremath{-}22}--{10}^{\ensuremath{-}21}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$. It was argued that a measurement process required to distinguish between the parent and the daughter nuclear states in such a short time would destroy the characteristics of the long-lived \ensuremath{\beta}-unstable nuclear state, thus fundamentally restricting the observability of \ensuremath{\beta} decay on a short timescale. Since the interaction of the nuclear state with the environment is almost inevitable, we have obtained the timescale of initial non-exponential decay for the nuclear \ensuremath{\beta} decay from an estimation of quantum decoherence time considering the atom of the decaying nucleus as a quantum detector. It has been found from such considerations that a decoherence timescale of \ensuremath{\beta} decay is on the order of ${10}^{\ensuremath{-}16}--{10}^{\ensuremath{-}15}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$ and the decay should remain reversible and non-exponential on this timescale. The possibilities of observing the effect of non-exponential decay in nuclear systems have been discussed.

Physical Review C•Nov 30, 2016•5 citations
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.
We consider a driven 2-level system with one level showing spontaneous decay to an otherwise uncoupled third level. Rabi transitions to the unstable level are strongly damped. This simple configuration can be used to demonstrate and to explore the quantum Zeno effect leading to a freezing of the system to the initial level. A comparison with repeated projection measurements is given. A treatment within a phenomenological theory of continuous measurements is sketched. The system visualizes the important role of null measurements (negative result measurements) and may serve as a good example for a truly continuous measurement. 1 1
The quantum-mechanical time evolution of an isolated \ensuremath{\beta}-unstable nuclear state in the context of certain models predicts that the square of the amplitude of the initial undecayed state becomes an approximately exponential function of time on a timescale on the order of ${10}^{\ensuremath{-}22}--{10}^{\ensuremath{-}21}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$. It was argued that a measurement process required to distinguish between the parent and the daughter nuclear states in such a short time would destroy the characteristics of the long-lived \ensuremath{\beta}-unstable nuclear state, thus fundamentally restricting the observability of \ensuremath{\beta} decay on a short timescale. Since the interaction of the nuclear state with the environment is almost inevitable, we have obtained the timescale of initial non-exponential decay for the nuclear \ensuremath{\beta} decay from an estimation of quantum decoherence time considering the atom of the decaying nucleus as a quantum detector. It has been found from such considerations that a decoherence timescale of \ensuremath{\beta} decay is on the order of ${10}^{\ensuremath{-}16}--{10}^{\ensuremath{-}15}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$ and the decay should remain reversible and non-exponential on this timescale. The possibilities of observing the effect of non-exponential decay in nuclear systems have been discussed.