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Journal of physics. A, Proceedings of the Physical Society. General

Some properties of coherent spin states

J. M. Radcliffe

Published May 1, 1971• 925 citations
Publisher

Abstract

Spin states analogous to the coherent states of the linear harmonic oscillator are defined and their properties discussed. They are used to discuss some simple problems (a single spin in a field, a spin wave, two spin 1 / 2 particles with Heisenberg coupling) and it is shown that their use may often give increased physical insight.
DOI
10.1088/0305-4470/4/3/009

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This chapter presents some basic aspects of coherent and squeezed states. Certain superpositions of energy Eigen functions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. The chapter obtains the Heisenberg uncertainty relation for the harmonic oscillator states and shows that it is a minimum for ground state. It defines coherent states and shows that they are the minimum uncertainty product states. Coherent states were constructed for many interesting potentials including infinite well and Poschl-Teller potentials and Morse potential. A particular quantum state of a spin system which most closely resembles a classical spin is termed as a spin coherent state. Use of squeezed states in quantum cryptography, quantum distribution of keys, extended coherence time, quantum engineering, teleportation, optics, detection of gravitational waves, quantum metrology and laser interferometers have been analyzed.
For the description of an assembly of two-level atoms, atomic coherent states can be defined which have properties analogous to those of the field coherent states. The analogy is not fortuitous, but is shown to be related to the group contraction of exponential operators based on the angular momentum algebra to exponential operators based on the harmonic-oscillator algebra. The derivation of the properties of the atomic coherent states is made easier by the use of a powerful disentangling theorem for exponential angular momentum operators. A complete labeling of the atomic states is developed and many of their properties are studied. In particular it is shown that the atomic coherent states are the quantum analogs of classical dipoles, and that they can be produced by classical fields.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.