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Journal of the Optical Society of America B

Decomposition formulas for su(1, 1) and su(2) Lie algebras and their applications in quantum optics

Masashi Ban

Published Aug 1, 1993• 127 citations
Publisher

Abstract

Generalized normal- and antinormal-order decomposition formulas are derived for exponential functions of the generators of su(1, 1) and su(2) Lie algebras. The expectation values of analytic functions of these generators are calculated in terms of the generalized decomposition formulas, allowing su(1, 1) and su(2) fluctuations and their squeezing properties to be discussed. The second-order correlation functions are also considered. It is shown that the Mach–Zehnder interferometer, which is described by su(2) Lie algebra, is equivalent to a three-dimensional rotation and is characterized by the quantum counterparts of the classical Euler angles.

DOI
10.1364/josab.10.001347

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Generalized normal- and antinormal-order decomposition formulas are derived for exponential functions of the generators of su(1, 1) and su(2) Lie algebras. The expectation values of analytic functions of these generators are calculated in terms of the generalized decomposition formulas, allowing su(1, 1) and su(2) fluctuations and their squeezing properties to be discussed. The second-order correlation functions are also considered. It is shown that the Mach–Zehnder interferometer, which is described by su(2) Lie algebra, is equivalent to a three-dimensional rotation and is characterized by the quantum counterparts of the classical Euler angles.
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.
Decomposition formulas of general exponential operators in a Banach algebra and in a Lie algebra are presented that yield a basis of Monte Carlo simulation of quantum systems. They are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena. A global approximation method of transient phenomena is also formulated on the basis of decomposition formulas. It is applied to the laser model as a simple example.
It is shown that a SU(1,1) algebra may be used to provide a unified description of the simple hamonic oscillator and the angular momentum algebras and a class of other semi-infinite algebras. A normal ordered representation of a Unitary operator UUU constructed from the generators of a SU(1,1) algebra, which is a generalisation of the Baker- Campbell - Hausdorff relation for Lie algebras, is given. It is shown that the normal ordered representatiion of UUU may be used to calculate expectation values which are functions of the parameters used to construct the operator. The functions so constructed satisfy certain recurrence relations and the entire set of functions may be interpreted in terms of diagrams similar to the Pascal triangle for binomial coefficients. Coherent states, squeezed states and rotation matrices of the angular momentum algebra emerge as special cases.
The theory of generalized coherent states associated with the SU(2) and SU(1,1) Lie algebras is discussed and applied in an investigation of possible reductions of fluctuations. In the framework of a system of N two-level atoms the squeezing of angular-momentum [SU(2)] fluctuations is exhibited for optical coherent transients involving the photon echo. The SU(1,1) fluctuations are discussed and established for general two-photon processes involving dynamical variables different from the creation and annihilation operators.