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Physical Review

Description of a Many-Atom System in Terms of Coherent Boson States

R. Bonifacio, Dae M. Kim, Marlan O. Scully

Published Nov 10, 1969• 45 citations
Publisher

Abstract

The correlated motion of an $N$-spin system has been described in terms of the coherent states of a harmonic oscillator. We have shown that this description of the system provides us with the closest analog to the classical limit. The macroscopic dipole operator in the coherent-state representation is shown to obey the Bloch equation in which each of the three components is simultaneously well defined.

DOI
10.1103/physrev.187.441

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The correlated motion of an NNN-spin system has been described in terms of the coherent states of a harmonic oscillator. We have shown that this description of the system provides us with the closest analog to the classical limit. The macroscopic dipole operator in the coherent-state representation is shown to obey the Bloch equation in which each of the three components is simultaneously well defined.
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.
We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.
In terms of the boson coherent state and a possible extension, three forms of wave packets are presented in the Schwinger boson representation. Each wave packet gives classical counterpart for a certain set of operators, for example, three components of the su(2)-spin. Furthermore, uncertainty relations which come from the boson coherent state and its extension are examined. The content discussed in this paper is, in a certain sense, an extension of that discussed in a paper presented recently by the present authors.
We study a many-body system consisting of a central harmonic oscillator linearly coupled to a reservoir of a large number of oscillators. To get the quantal description of the central oscillator we use normal-ordering operators and the coherent-state representation in order to solve the Schr\"odinger equation for the complete system. Then, it is shown that the wave function for the total system allows one to obtain the following results: (i) The expectation value of the coordinate of the central oscillator is the same as the classical solution for the damped harmonic oscillator and (ii) the energy of the central oscillator decays towards the correct zero-point energy.