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IEEE Conference on Decision and Control (CDC)

Quantum control using Lie group decompositions

S. G. Schirmer

Published Nov 13, 2002• 4 citations
Publisher

Abstract

Lie group decompositions of a desired unitary evolution operator are employed to find simple control schemes to achieve population transfers for systems initially in the ground state as well as complete inversions of the ensemble populations for systems whose initial state is an ensemble of energy eigenstates. Both completely controllable and pure-state controllable systems are considered and it is shown that the control objective can be achieved using realistic square wave pulses.

DOI
10.1109/.2001.980116

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Lie group decompositions of a desired unitary evolution operator are employed to find simple control schemes to achieve population transfers for systems initially in the ground state as well as complete inversions of the ensemble populations for systems whose initial state is an ensemble of energy eigenstates. Both completely controllable and pure-state controllable systems are considered and it is shown that the control objective can be achieved using realistic square wave pulses.
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We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique, it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.
The purpose of controlling a quantum system is to steer the system from an initial state to a target state. Such a control problem can be changed into the problem of generating the desired unitary evolution operators by using an appropriate decomposition. By means of the desired state's exponential factor product theorem in the right-invariant system, the desired unitary evolution operators of a quantum system can be obtained. This paper investigates the functions and features of the Magnus decomposition, the Wei-Norman decomposition, the Lie Group decomposition and the Cartan decomposition contrastively and respectively. Properties and realization performance of those decompositions are analyzed, too. Each type of decomposition applied to different situations is also given.
A new notion of controllability for quantum systems that takes advantage of the linear superposition of quantum states is introduced. We call such notion von Neumann controllabilty and it is shown that it is strictly weaker than the usual notion of pure state and operator controlability. We provide a simple and effective characterization of it by using tools from the theory of unitary representations of Lie groups. In this sense we are able to approach the problem of control of quantum states from a new perspective, that of the theory of unitary representations of Lie groups. A few examples of physical interest and the particular instances of compact and nilpotent dynamical Lie groups are discussed.