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arXiv:1711.01071

Quantum Zeno Effect, Kapitsa Pendulum and Spinning Top Principle. Comparative Analysis

V. A. Buts

Published Nov 3, 2017• 1 citation
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Abstract

Comparative analysis of three stabilization mechanisms of unstable states of physical systems is presented in this review. These mechanisms are: the quantum Zeno effect, the stabilization of unstable states in an external fast oscillating field (i.e. the Kapitza pendulum), and the mechanism named as the spinning top mechanism. The common features of these mechanisms, as well as the differences between them, are analyzed in the paper. In particular, it is shown that the stabilization of quantum systems is possible without involvement of such concept as the collapse of wave function. For stabilization it is sufficient to have a stabilizing radiation flow with the Rabi frequency of transitions exceeding some frequency. This frequency is inversely proportional to the lifetime of the state under stabilization. It is shown that the Top principle allows stabilizing unstable systems using affecting only those states to which these systems must go over. It is shown that stabilization of unstable states by impact of rapidly oscillating forces occurs by non-self-consistent exposure, i.e. the dynamics of stabilizing field is independent on the dynamics of the stabilized state. Stabilization using the spinning top principle involves self-consistent forces, and thus, in many cases can be the most effective mechanism of stabilization.

DOI
10.48550/arxiv.1711.01071

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Comparative analysis of three stabilization mechanisms of unstable states of physical systems is presented in this review. These mechanisms are: the quantum Zeno effect, the stabilization of unstable states in an external fast oscillating field (i.e. the Kapitza pendulum), and the mechanism named as the spinning top mechanism. The common features of these mechanisms, as well as the differences between them, are analyzed in the paper. In particular, it is shown that the stabilization of quantum systems is possible without involvement of such concept as the collapse of wave function. For stabilization it is sufficient to have a stabilizing radiation flow with the Rabi frequency of transitions exceeding some frequency. This frequency is inversely proportional to the lifetime of the state under stabilization. It is shown that the Top principle allows stabilizing unstable systems using affecting only those states to which these systems must go over. It is shown that stabilization of unstable states by impact of rapidly oscillating forces occurs by non-self-consistent exposure, i.e. the dynamics of stabilizing field is independent on the dynamics of the stabilized state. Stabilization using the spinning top principle involves self-consistent forces, and thus, in many cases can be the most effective mechanism of stabilization.
Comparative analysis of three stabilization mechanisms of unstable states of physical systems is presented in this review. These mechanisms are: the quantum Zeno effect, the stabilization of unstable states in an external fast oscillating field (at the example of Kapitza pendulum), and the algorithm called as the principle of whirligig. The common features of these mechanisms, as well as the differences between them, are defined in the paper. In particular, it is shown that the stabilization of quantum systems is possible without involvement of such a notion as the collapse of the wave function. For stabilization there is enough to have such a stabilizing radiation flow when the Rabi frequency of transitions does exceed some frequency. This particular frequency is a magnitude inverse-ly proportional to the lifetime of the state under stabilization. It is shown that stabilization of unstable states by impact of rapidly oscillating forces occurs by non-self-consistent exposure, i.e. the dynamics of stabilizing field is independent of the dynamics of the stabilized state. Stabilization with the principle of whirligig does occur by self-consistent forces, and thus, in many cases stabilization by the use of the principle of whirligig can be more effective.
We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate.
We analyze the Zeno phenomenon in quantum field theory. The decay of an unstable system can be modified by changing the time interval between successive measurements (or by varying the coupling to an external system that plays the role of measuring apparatus). We speak of quantum Zeno effect if the decay is slowed and of inverse quantum Zeno (or Heraclitus) effect if it is accelerated. The analysis of the transition between these two regimes requires close scrutiny of the features of the interaction Hamiltonian. We look in detail at quantum field theoretical models of the Lee type.
A metastable atomic level can be rendered unstable in a controllable way by coupling it to a decaying state. In this work we carry out a full dynamical analysis of the Zeno effect in this kind of unstable systems, comparing it to the inhibition of purely coherent Rabi oscillations. Simple and experimentally feasible measuring strategies involving three atomic levels are considered. It is shown that this induced decay is actually an example of a partial Zeno effect so that the observed evolution results from the competition of two Zeno effects. We also show that a three-level scheme can display both coherent, incoherent, and anti-Zeno effects.
Frequent Measurements on an unstable particle located at observable initial state freeze the particle on this state, known as quantum Zeno effect [1-14]. Measurements on an observable subspace further open the prelude of quantum Zeno dynamics [15-17]. These phenomena affect the results of quantum measurement that has been widely used in quantum information and quantum computation [18-21]. However, this common argument is insufficient when the initial state is coupled with a continuum reservoir. In such an irreversible system, intrinsic decay property destroys the frozen behavior. Although it has been proven that the decay rate of the initial state will be affected by measurements [7], the knowledge of detailed mechanisms for this measurement-dependent decay is still limited. In this work, we found based on three-subsystem perturbation theory that the indirect correlation between a continuum reservoir and a measurement apparatus through the interaction with a quantum system causes a Rabi-like inter-modulation among these subsystems. The indirect correlation effect from this combination drives the system into quantum Zeno and reactivation regions, which are dependent upon the Rabi strength, pulse duty, and its repetition rate. We also found that the dynamical changes of decay rate are quite different in continuous and pulsed measurements, and, thus, significantly affect the dynamics of the system.