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arXiv:0908.1301

Quantum Hamlet Effect

Vladan Panković

Published Aug 10, 2009• 1 citation
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Abstract

In this work, by use of a formalism similar to formalism of the quantum Zeno effect (decrease of the decay probability of an unstable quantum system by frequent measurements) and quantum anti-Zeno effect (increase of the decay probability of an unstable quantum system by frequent measurements), we introduce so-called quantum Hamlet effect. It represents a complete destruction of the quantum predictions on the decay probability of an unstable quantum system by frequent measurement. Precisely, by means of some especial, correctly defined, frequent measurements, decay probability of an unstable quantum system can behave as a divergent series without any definite value. In this way there is quantum mechanically completely unsolvable “Hamlet dilemma”, to decay or not to decay. As it is well-known [1] quantum Zeno effect describes decrease of the

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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.

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QUANTUM HAMLET EFFECT- A NEW EXAMPLE

Simply speaking quantum Zeno effect for an unstable quantum system represents total decay probability decrease by frequent decay detections.Analogously simply speaking quantum anti-Zeno effect for an unstable quantum system represents total no-decay probability decrease by frequent decay detections.Also, some authors considered optical, i.e. polarization detection analogy of quantum Zeno and anti-Zeno effects.In this work we consider basic standard quantum mechanical concepts in quantum Zeno and anti-Zeno effect.We use simplest (Malus law) form of mentioned ideas on the photon polarization detection analogy of quantum Zeno and anti-Zeno effects and we suggest some new situations (non-equivalent time durations of the quantum dynamical evolutions between different frequent detections).Correspondingly we define new quantum effects, quantum Hamlet effect (when, by frequent detections, final total polarization angle and state converge but total probability for appearance of this state diverges) and quantum anti-Hamlet effect (when, by frequent detections, final total polarization angle and state diverge but total probability for appearance of this state converges).Also, we consider a rough classical mechanical analogy of Zeno effect.Finally we consider frequent interference of single photon at beam splitters and obtain for different values of reflection and transmission coefficients analogies of quantum Zeno and anti-Zeno effect as well as analogy of usual exponentially decreasing decay law.

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In this work, by use of a formalism similar to formalism of the quantum Zeno effect (decrease of the decay probability of an unstable quantum system by frequent measurements) and quantum anti-Zeno effect (increase of the decay probability of an unstable quantum system by frequent measurements), we introduce so-called quantum Hamlet effect. It represents a complete destruction of the quantum predictions on the decay probability of an unstable quantum system by frequent measurement. Precisely, by means of some especial, correctly defined, frequent measurements, decay probability of an unstable quantum system can behave as a divergent series without any definite value. In this way there is quantum mechanically completely unsolvable “Hamlet dilemma”, to decay or not to decay. As it is well-known [1] quantum Zeno effect describes decrease of the
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.
Simply speaking quantum Zeno effect for an unstable quantum system represents total decay probability decrease by frequent decay detections.Analogously simply speaking quantum anti-Zeno effect for an unstable quantum system represents total no-decay probability decrease by frequent decay detections.Also, some authors considered optical, i.e. polarization detection analogy of quantum Zeno and anti-Zeno effects.In this work we consider basic standard quantum mechanical concepts in quantum Zeno and anti-Zeno effect.We use simplest (Malus law) form of mentioned ideas on the photon polarization detection analogy of quantum Zeno and anti-Zeno effects and we suggest some new situations (non-equivalent time durations of the quantum dynamical evolutions between different frequent detections).Correspondingly we define new quantum effects, quantum Hamlet effect (when, by frequent detections, final total polarization angle and state converge but total probability for appearance of this state diverges) and quantum anti-Hamlet effect (when, by frequent detections, final total polarization angle and state diverge but total probability for appearance of this state converges).Also, we consider a rough classical mechanical analogy of Zeno effect.Finally we consider frequent interference of single photon at beam splitters and obtain for different values of reflection and transmission coefficients analogies of quantum Zeno and anti-Zeno effect as well as analogy of usual exponentially decreasing decay law.
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.
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.
The quantum Zeno effect consists in the hindrance of the evolution of a quantum system that is very frequently monitored and found to be in its initial state at every single measurement. On the basis of the correct formula for the survival probability, i.e. the probability of finding the system in its initial state at every single measurement, we critically analyze a recent proposal and experimental test, that make use of an oscillating system. PACS: 03.65.Bz, 42.50.-p 1 The seminal formulation of the quantum Zeno effect, due to Misra and Sudarshan [1], deals with the probability of observing an unstable system in its initial state throughout a time interval ∆ = [0, t]. The purpose of this note is to point out that the quantum Zeno effect has not been experimentally observed, yet, in its original formulation. Indeed, we shall argue that the interesting proposal by Cook [2], that makes use of a two-level system undergoing Rabi oscillations, as well as the beautiful experiment performed by Itano et al. [3], investigate the probability of finding the