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Journal of Physics A Mathematical Nuclear and General

On coherent spin states

R. Holtz, J Hanus

Published Mar 1, 1974• 14 citations
Publisher

Abstract

Some properties of so called coherent spin states, analogous to the coherent states of the harmonic oscillator, are discussed. Polynomials of order 2S play an important role. The representation of spin operators in the space of coherent spin states is found. This representation is used to study the rotation operator.
DOI
10.1088/0305-4470/7/4/001

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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.
Introduction In our discussion of the classical electromagnetic field we found it convenient to describe the field by a complex amplitude, both in the frequency and in the time domains. The complex representation is convenient partly because it contains information about the magnitude and about the phase of the electromagnetic disturbance, and partly because of its analytic properties. These features turn out to be particularly useful for the description of the optical coherence properties of the field. We shall see that an analogous quantum state of the field exists, leading to an interesting representation that is also particularly useful for the treatment of optical coherence. This coherent-state representation leads to a close correspondence between the quantum and classical correlation functions. The coherent states of the field come as close as possible to being classical states of definite complex amplitude. We shall find that coherent states turn out to be particularly appropriate for the description of the electromagnetic fields generated by coherent sources, like lasers and parametric oscillators; indeed it turns out that the field produced by any deterministic current source is in a coherent state. In the following sections we shall examine some of the properties of coherent states, and then go on to discuss representations based on these states. Coherent states were first discovered in connection with the quantum harmonic oscillator by Schrödinger (1926), who referred to them as states of minimum uncertainty product.
Hermite polynomials with many variables and many indices play a crucial role within the framework of phase-space formulation of classical or quantum mechanics. A generalized operational formalism useful to handle these polynomials is discussed and also a new set of creation–annihilation operators associated to the phase-space harmonic oscillator orthogonal functions is introduced. The Fourier transform of these orthogonal functions is studied by discussing the analogies with the ordinary case. Finally, the integral representation of the generalized Hermite polynomials and a simple technique to deal with the generalized heat equation are discussed.