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Physical Review A

Dynamic Lie-algebra structure of a quantum system and the Aharonov-Anandan phase

Shi-Min Cui

Published Jan 1, 1993• 2 citations
Publisher

Abstract

The dynamic Lie-algebra structure decomposition of the time-dependent Schr\"odinger equation for a quantum system is analyzed and the results are used to study the Aharonov-Anandan phase. The expression for Aharonov-Anandan phase is given, which is shown to be useful by the illustration of some examples and to be applicable to rather general systems.

DOI
10.1103/physreva.47.690

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The dynamic Lie-algebra structure decomposition of the time-dependent Schr"odinger equation for a quantum system is analyzed and the results are used to study the Aharonov-Anandan phase. The expression for Aharonov-Anandan phase is given, which is shown to be useful by the illustration of some examples and to be applicable to rather general systems.
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Quantum technologies rely on the control of quantum systems at the level of individual quanta. Mathematically, this control is described by Hamiltonian or Liouvillian evolution, requiring the application of various techniques to solve the resulting dynamic equations. Here, we present a tutorial for how the quantum dynamics of systems can be solved using a Lie-algebra decoupling method. The approach involves identifying a Lie algebra that governs the dynamics of the system, enabling the derivation of differential equations to solve the Schrödinger equation. As background, we include an overview of Lie groups and Lie algebras aimed at a general-physicist audience. We then prove the Lie-algebra decoupling theorem and apply it to both closed and open dynamics. The results represent a broad methodology to find the dynamics of quantum systems with applications across many fields of modern quantum research.
In this paper we generalize Wang's approach [J. Phys. A 20, 5041 (1987)] and investigate the algebraic structure of the Schr\"odinger equation associated with a general driven time-dependent oscillator. Using the Lie-algebraic technique we obtain an exact form of the time-evolution operator which, in turn, enables us to derive the propagator of the system readily. Since the propagator is for the most general time-dependent oscillator, results for any special case can be easily deduced from it. These results will be useful for future studies in quantum optics as well as in atomic and molecular physics.