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USE OF THE WHIRLIGIG PRINCIPLE FOR STABILIZING THE INITIAL STATES OF SOME CLASSIC AND QUANTUM SYSTEMS

V. A. Buts

Published Sep 21, 2020• 1 citation
Publisher

Abstract

It is shown that the whirligig principle can be used for stabilization of the initial states of some classical and quantum systems. This feature of the whirligig principle is demonstrated by simple examples. The most important result of this work is the proof of the fact that the stabilization of the excited states of quantum systems can be realized by acting not on the quantum system itself, but by acting on the states into which the system must go. Potentially, this result can be used to stabilize excited nuclear systems.

DOI
10.46813/2020-129-078

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It is shown that the whirligig principle can be used for stabilization of the initial states of some classical and quantum systems. This feature of the whirligig principle is demonstrated by simple examples. The most important result of this work is the proof of the fact that the stabilization of the excited states of quantum systems can be realized by acting not on the quantum system itself, but by acting on the states into which the system must go. Potentially, this result can be used to stabilize excited nuclear systems.
The mechanism allowing to stabilize of a state of quantum systems is considered. And, the initial condition can correspond both for excited state and for not excited, stationary state. The considered mechanism for the first time was offered for the excited states, and has received the name as quantum whirligig (QWE). In this work the close connection of the considered mechanism with Zeno effect is shown. The considerations are stated, that many experimental results, which are interpreted as observation of Zeno effect, apparently, correspond to QWE.
As compared to the fact that the control laws must be performed on the systems eventually, in the quantum control theory, most of the deliberately designed control laws and their guaranteed satisfactory performances are based on the models of the systems. Therefor, one natural and central problem is how to ensure the satisfactory performances of the designed control laws when they are being performed on the systems. In this paper, we will exhibit one simple yet efficient strategy, the separation principle, by focusing on a state stabilization problem. Specifically, one first designs the control strategy based on a model of the system, and then realizes the control laws to be performed on the real system by open-loop control technology to deal with the differences between the system and the established model. It will be shown that the quantum characteristics, measurement-induced-state-transfer and the tensor product structure of the coupling systems, play essential roles in the total control process.
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 consider classical normal modes and noninteracting bosonic excitations in disordered systems. We emphasize generic aspects of such problems and parallels with disordered, noninteracting systems of fermions, and discuss in particular the relevance for bosonic excitations of symmetry classes known in the fermionic context. We also stress important differences between bosonic and fermionic problems. One of these follows from the fact that ground-state stability of a system requires all bosonic excitation energy levels to be positive, while stability in systems of noninteracting fermions is ensured by the exclusion principle, whatever the single-particle energies. As a consequence, simple models of uncorrelated disorder are less useful for bosonic systems than for fermionic ones, and it is generally important to study the excitation spectrum in conjunction with the problem of constructing a disorder-dependent ground state: we show how a mapping to an operator with chiral symmetry provides a useful tool for doing this. A second difference involves the distinction for bosonic systems between excitations which are Goldstone modes and those which are not. In the case of Goldstone modes we review established results illustrating the fact that disorder decouples from excitations in the low-frequency limit, above a critical dimension dc,{d}_{c},dc​, which in different circumstances takes the values dc=2{d}_{c}=2dc​=2 and dc=0.{d}_{c}=0.dc​=0. For bosonic excitations which are not Goldstone modes, we argue that an excitation density varying with frequency as $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{4}$ is a universal feature in systems with ground states that depend on the disorder realization. We illustrate our conclusions with extensive analytical and some numerical calculations for a variety of models in one dimension.