Lie-algebra methods in quantum optics: The Liouville-space formulation

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Abstract
Lie-algebra methods for investigating quantum optical systems are presented within the framework of the Liouville-space formulation. The generalized decomposition formulas for exponential functions of the generators of su(2) and su(1,1) Lie algebras are derived and their expectation values are calculated for typical states in quantum optics. The general procedure for using Lie algebras in the Liouville space to treat quantum optical processes is given in terms of generalized decomposition formulas and their use is demonstrated by calculating the absorption line shape and photon echo signal in a localized electron-phonon system. It is also shown that the photon-counting probability can be calculated by using the su(1,1) Lie algebra and that the electron-counting probability can be calculated by using the su(2) Lie algebra. The su(1,1) Lie algebra is also used to investigate a quantum-nondemolition measurement of photon number in the four-wave-mixing model.