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CRC Press eBooks

Coherent state realizations of su(n + 1) in terms of subgroup functions

Hubert de Guise, Marco Bertoia

Published Nov 30, 2003• 0 citations
Publisher

Abstract

We present a new kind of coherent state realizations, where generators of the algebra su ( n + 1) (or, more properly, its complex extension) are expressed in terms of simple actions and subgroup functions. Phase operators and other potential applications to quantum optics are briefly discussed.
DOI
10.1201/9781482269079-99

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We start with the Heisenberg–Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1, 1) algebras and their coherent states. From there we go on describing the binomial state and its extensions as realizations of the SU(2) group. This is followed by considering the negative binomial states, and some squeezed states as realizations of the SU(1, 1) group. Generation schemes based on physical systems are mentioned for some of these states.
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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We introduce and study the properties of a class of coherent states for the group SU(1,1)\ifmmode\times\else\texttimes\fi{}SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We also derive a phase space representation based on pair coherent states rather than the standard harmonic-oscillator coherent states. We discuss the utility of the resulting ``bi-pair coherent states'' in the context of four-mode interactions in quantum optics.