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arXiv:0707.2624 · AIP conference proceedings

Analytic Approach for Controlling Realistic Quantum Chaotic Systems

Toshiya Takami, Hiroshi Fujisaki, Theodore E. Simos, George Maroulis

Published Jan 1, 2007• 1 citation
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Abstract

An analytic approach for controlling quantum states, which was originally applied to fully random matrix systems [T. Takami and H. Fujisaki, Phys. Rev. E 75, 036219 (2007)], is extended to deal with more realistic quantum systems with a banded random matrix (BRM). The validity of the new analytic field is confirmed by directly solving the Schrödinger equation with a BRM interaction. We find a threshold of the width of the BRM for the quantum control to be successful.

DOI
10.1063/1.2836218

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An analytic approach for controlling quantum states, which was originally applied to fully random matrix systems [T. Takami and H. Fujisaki, Phys. Rev. E 75, 036219 (2007)], is extended to deal with more realistic quantum systems with a banded random matrix (BRM). The validity of the new analytic field is confirmed by directly solving the Schrödinger equation with a BRM interaction. We find a threshold of the width of the BRM for the quantum control to be successful.
The rotating-wave approximation and its validity in multi-state quantum systems are studied through analytic approaches. Their applicability is also verified from the viewpoint of generic states by the use of direct numerical integrations of the Schrödinger equation. First, we introduce an extension of the rotating-wave approximation for multi-state systems. Under an assumption that a smooth transition is induced by the optimal field, we obtain three types of analytic control fields to demonstrate their validity and deficiency for generic systems represented by random matrices. Through the comparison, we conclude that the analytic field based on our coarse-grained approach outperforms the other ones for generic quantum systems with a large number of states. Finally, the further extension of the analytic field is introduced for realistic chaotic systems and its validity is shown in banded random matrix systems.
We propose a coarse-grained picture to control ``complex'' quantum dynamics, i.e., multi-level-multi-level transition with a random interaction. Assuming that optimally controlled dynamics can be described as a Rabi-like oscillation between an initial and final state, we derive an analytic optimal field as a solution to optimal control theory. For random matrix systems, we numerically confirm that the analytic optimal field steers an initial state to a target state which both contains many eigenstates.
Quantum-enhanced chaos represents a paradigm shift in our understanding and manipulation of chaotic systems. This research delves into the design of a novel quantum algorithm, termed the "Quantum Resonance" method, to achieve precise control over chaotic dynamics. The core claim is that this algorithm, leveraging quantum fluctuations, can guide and shape chaotic systems, moving beyond the limitations of classical control techniques. We will explore the theoretical underpinnings of the Quantum Resonance method, detailing its operational principles, potential advantages, and challenges in practical implementation. This work proposes a framework for harnessing the unique properties of quantum mechanics to achieve unprecedented levels of control over chaotic systems.
Abstract We substantially extend our relaxation theory for perturbed many-body quantum systems from ((2020) Phys. Rev. Lett. 124 120602) by establishing an analytical prediction for the time-dependent observable expectation values which depends on only two characteristic parameters of the perturbation operator: its overall strength and its range or band width. Compared to the previous theory, a significantly larger range of perturbation strengths is covered. The results are obtained within a typicality framework by solving the pertinent random matrix problem exactly for a certain class of banded perturbations and by demonstrating the (approximative) universality of these solutions, which allows us to adopt them to considerably more general classes of perturbations. We also verify the prediction by comparison with several numerical examples.
This paper presents a novel approach to modeling and controlling quantum chaotic systems. The core idea revolves around leveraging the principles of quantum dynamics to construct accurate mathematical models of chaotic systems, followed by the application of quantum control techniques for precise manipulation and prediction. The methodology addresses a critical gap in the current research landscape, which is the nascent stage of quantum chaos studies. This work lays the groundwork for future advancements in this field, offering a systematic framework for both modeling and control. Specifically, we outline a model construction process based on the Schrödinger equation and detail control strategies utilizing quantum interference and feedback mechanisms. The presented framework has potential applications in areas such as quantum simulation, secure communication, and precision measurement.