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Journal of Modern Optics

Quantum Phase Angles and su(∞)

Demosthenes Ellinas

Published Dec 1, 1991• 14 citations
Publisher

Abstract

The polar decomposition of the su(2) algebra leads to unitary phase operators which do not close to an algebra with the number operator. It is shown that phase and number operators can be embedded into a larger su(2j + 1) algebra with trigonometric structure constants. In the contraction limit where we pass from the su(2) to the oscillator algebra, the embedding algebra for the phase operators becomes su(∞). The coherent states realization of the su(∞) algebra and its relation to the q-deformed oscillator algebra is briefly discussed.

DOI
10.1080/09500349114552541

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The polar decomposition of the su(2) algebra leads to unitary phase operators which do not close to an algebra with the number operator. It is shown that phase and number operators can be embedded into a larger su(2j + 1) algebra with trigonometric structure constants. In the contraction limit where we pass from the su(2) to the oscillator algebra, the embedding algebra for the phase operators becomes su(∞). The coherent states realization of the su(∞) algebra and its relation to the q-deformed oscillator algebra is briefly discussed.
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A new solution is proposed to the longstanding problem of describing the quantum phase of a harmonic oscillator. In terms of an 'exponential phase operator', defined by a new 'polar decomposition' of the quantized amplitude of the oscillator, a regular phase operator is constructed in the Hilbert–Fock space as a strongly convergent power series. It is shown that the eigenstates of the new 'exponential phase operator' are SU(1,1) coherent states associated to the Holstein–Primakoff realization. In terms of these eigenstates the diagonal representation of phase densities and a generalized spectral resolution of the regular phase operator are derived, which are very well suited to our intuitive pictures of classical phase-related quantities.