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Physical Review A

SU(1,1), SU(2), and SU(3) coherence-preserving Hamiltonians and time-ordering techniques

G. Dattoli, P. Di Lazzaro, A. Torre

Published Feb 1, 1987• 32 citations
Publisher

Abstract

The Wei-Norman ordering technique is utilized to present a unified approach to the time ordering for SU(2), SU(1,1), and SU(3) coherence-preserving Hamiltonians. It is shown that the characteristic equations of the ordering method can be cast in the form of a generalized Bloch equation in an SU(2), SU(1,1), and SU(3) space. The laws of conservation, linked to the Casimir invariants of the algebra, are also directly derived from the Wei-Norman characteristic equations.

DOI
10.1103/physreva.35.1582

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The Wei-Norman ordering technique is utilized to present a unified approach to the time ordering for SU(2), SU(1,1), and SU(3) coherence-preserving Hamiltonians. It is shown that the characteristic equations of the ordering method can be cast in the form of a generalized Bloch equation in an SU(2), SU(1,1), and SU(3) space. The laws of conservation, linked to the Casimir invariants of the algebra, are also directly derived from the Wei-Norman characteristic equations.
The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms of Euler angles. The many applications are postponed to Part II of the paper.
In this work we discuss the problems underlying the time evolution of states driven by SU(2) and SU(1,1) coherence-preserving Hamiltonians, using the Wei-Norman algebraic procedure. We also solve the problem of finding the explicit time evolution of a quantum system whose Hamiltonian is a time-dependent linear combination of the SU(1,1) group generators and those of the Weyl-Heisenberg algebra. The relevance of this Hamiltonian to the generation of antibunched radiation in a laser-plasma scattering experiment is also briefly analyzed.