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IEEE Conference on Decision and Control (CDC)

Control of Heisenberg spin systems; Lie algebraic decompositions and action-angle variables

Umesh Vaidya, Deanna M. D’Alessandro, Igor Mezić

Published Apr 23, 2004• 12 citations
Publisher

Abstract

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.

DOI
10.1109/cdc.2003.1271804

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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.
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In this paper, we present a controllability analysis of the quantum Ising periodic chain of n spin 1/2 particles where the interpolating parameter between the two Hamiltonians plays the role of the control. A fundamental result in the control theory of quantum systems states that the set of achievable evolutions is (dense in) the Lie group corresponding to the Lie algebra generated by the Hamiltonians of the system. Such a dynamical Lie algebra therefore characterizes all the state transitions available for a given system. For the Ising spin periodic chain we characterize such a dynamical Lie algebra and therefore the set of all reachable states. In particular, we prove that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of su(2^n) which is a direct sum of a two dimensional center and a (3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed matrix. We display the basis for each of these Lie subalgebras. Therefore the problem of control for the Ising spin periodic chain is, modulo the two dimensional center, a problem of simultaneous control of n-1 spin 1/2 particles. In the process of proving this result, we develop some tools which are of general interest for the controllability analysis of quantum systems with symmetry.
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Controllability of dynamical systems is a fundamental problem for both classical control as well as quantum control.This paper focuses on investigating the controllability Lie algebra of single-spin and two spin quantum systems.Firstly,based on the equivalent controllability conditions of quantum systems,some general necessary and sufficient conditions related to the basis coefficients are obtained by computing Lie brackets of Hamiltonian operators for single-spin quantum systems.Secondly,the matrix basis of Lie algebra su(4) is generated by Cartan decomposition.Using the matrix basis and employing the controllability theorem of quantum system,a calculation method of multiple Lie brackets for the Hamiltonian operators and a controllability criterion of quantum systems are also put forward in this article.