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Control theory & applications

Quantum system model of interaction quantum system and its physical control procedure

Shuang Cong

Published Jan 1, 2006• 1 citation
OpenAlex

Abstract

Based on the interation between particles of quantum system and the possible geometric control to be applied,a mathematical model of two spin 1/2 particle system with interaction is built,whose variables are varying in the Lie groups of SU(4).The physical control procedure of the quantum system with interaction is described in detail.This work is a preparation for further research on controllability,manipulation and feedback control of quantum system.

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Based on the interation between particles of quantum system and the possible geometric control to be applied,a mathematical model of two spin 1/2 particle system with interaction is built,whose variables are varying in the Lie groups of SU(4).The physical control procedure of the quantum system with interaction is described in detail.This work is a preparation for further research on controllability,manipulation and feedback control of quantum system.
This paper presents a novel approach to modeling dynamic quantum control scenarios involving nonlinear quantum systems. The core claim is the establishment of a model capable of accurately describing the behavior of such systems under dynamic control. This model is built upon the introduction of a nonlinear Schrödinger equation, coupled with principles from dynamic control theory. The methodology allows for detailed analysis and manipulation of nonlinear quantum systems, expanding the theoretical foundation for quantum control and manipulation research. We explore the challenges and potential solutions in designing control strategies for systems exhibiting complex, nonlinear interactions. The presented framework offers a foundation for further research into advanced quantum control techniques. The key mathematical components are detailed within the subsequent sections, including the nonlinear Schrödinger equation, control vector formulation, and analysis of system dynamics. The focus is on providing a rigorous mathematical framework rather than a specific experimental implementation.
Quantum systems whose dynamics depends on one or more control functions will be named quantum control systems. In a typical situation the objects of control are charged particles while the control functions are appropriately shaped electromagnetic fields. Therefore, to derive the models, in this chapter we describe elements of the theory of field-particle interaction. We start with the classical theory of electrodynamics and show how to obtain the quantum mechanical description through a process called canonical quantization. We present physical examples of (finite dimensional) quantum control systems of interest in applications and we highlight the fact that such systems have a common mathematical structure, as bilinear systems whose evolution is determined by an element of the Lie group of unitary matrices. In a parallel to classical control theory, we identify a state, an input, and an output in our description of quantum control systems.
In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.
In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.
In this paper, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution operator as well as the state of the system. We establish the connections among these different notions as well as methods to verify controllability. The paper also contains results on the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove that, for the system of two interacting spin 1/2 particles, not every state transfer can be obtained in arbitrary small time.
In this paper, we consider the control of two coupled spin 1/2 particles that interact via Heisenberg interaction. We decompose the problem into a sequence of subproblems on a lower dimensional Lie group whose Lie algebra is the direct sum of su(2) and a two dimensional Abelian subalgebra. The associated lower dimensional control problems amount to controlling a two level system. To this aim, we derive a control algorithm which, from an energy function, calculates a nominal feedback law. The results guarantee convergence of the state from any initial condition to the desired final configuration.