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Journal of Physics Conference Series

Non-entire Functions of Creation and Annihilation Operators and Their Relation to Phase Operator

M Davidović, D. Arsenović, Dragomir M Davidović

Published Apr 1, 2006• 1 citation
Publisher

Abstract

On the coherent states |α⟩ any entire function of creation and annihilation operators may be defined. We show that it is not the case for non-entire functions. Use of |α⟩⟨α| as identity operator for a non-entire function may lead to contradictory results. On the example of the phase operator we show how these possible contradictions may be avoided.

DOI
10.1088/1742-6596/36/1/009

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On the coherent states |α⟩ any entire function of creation and annihilation operators may be defined. We show that it is not the case for non-entire functions. Use of |α⟩⟨α| as identity operator for a non-entire function may lead to contradictory results. On the example of the phase operator we show how these possible contradictions may be avoided.
Generalized coherent states which are associated with a generalization of the harmonic oscillator commutation relation are investigated. It is shown that these states form an overcomplete basis in a Hilbert space of analytic functions. The generalized creation and annihilation operators are bounded except in a limit in which they reduce to the usual boson creation and annihilation operators. In this limit the Hilbert space of analytic functions reduces to the Bargmann–Segal Hilbert space of entire functions and in another limit it reduces to the Hardy–Lebesgue space.
The elementary quantum theory of quasi-classical states for mechanical harmonic oscillators is applied to the radiation oscillators describing light, and thus used to define quasi-classical states of light. These states are called “coherent”, because the expectation value of any normally ordered product of creation and annihilation operators factorizes for them. A theoretical discussion of possible sources for coherent states is followed by experimental evidence for their existence. The mathematical properties of coherent states, e.g., nonorthogonality and overcompleteness, are explained in order to use them to represent states and operators. These representations provide a suitable replacement for the classical phase space description of light, which is forbidden by the uncertainty principle. The quantum phase space approach leads to the definition of the Wigner distribution, the Q-function, the P-function, and Gaussian mixed states.