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Journal of Advanced Physics

Quantum Hamlet Effect—A New Example

Vladan Panković

Published May 13, 2016• 0 citations

Abstract

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.

DOI: 10.1166/jap.2016.1275

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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 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.

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.

In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such non-selective measurements. Importantly, we show that there is a difference between using non-selective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems, and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective measurements can differ very significantly as compared to the repeated selective measurement scenario.

The inhibition of the decay of a quantum system by frequent measurements is known as quantum Zeno effect. Beyond the limit of projective measurements, the interplay between the unitary dynamics of the system and the coupling to a measurement apparatus becomes relevant. We explore this interplay by considering a quantum particle moving on a one-dimensional chain. A local measurement by coupling to an apparatus with a two-dimensional Hilbert space detects the presence of the particle on a specific chain site. The decay of the population is studied analytically for a two-site chain and numerically for a larger system as a function of the measurement time and the time between subsequent measurements. Particular attention is given to the shift of the energy of the measured site due to the coupling to the apparatus. The decay of the initial population can be hindered or accelerated, depending on the chosen system and the coupling parameters.