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Physical Review Letters

Dynamical Meaning of Quantum Adiabatic Phase: The Case of a Noncanonical System

Hiroshi Kuratsuji, Shinji Iida

Published Mar 10, 1986• 41 citations
Publisher

Abstract

A dynamical meaning is given for the phase $\ensuremath{\Gamma}$ recently found in a quantum adiabatic process. A noncanonical system, specifically a spin system coupled with a proper intrinsic system, is considered by use of a path integral in a SU (2) coherent-state representation. It is shown that $\ensuremath{\Gamma}$ appears as an addition of topological nature to the conventional action function, which leads to a novel form of the semiclassical quantization rule including the phase $\ensuremath{\Gamma}$.

DOI
10.1103/physrevlett.56.1003

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A dynamical meaning is given for the phase \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ recently found in a quantum adiabatic process. A noncanonical system, specifically a spin system coupled with a proper intrinsic system, is considered by use of a path integral in a SU (2) coherent-state representation. It is shown that \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ appears as an addition of topological nature to the conventional action function, which leads to a novel form of the semiclassical quantization rule including the phase \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ.
Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.
The purpose of this paper is to give a somewhat expanded argument of the concept of the canonical phase recently investigated. This newly identified phase is considered to be a nonintegrable phase defined over the generalized phase space which is regarded as an integral realization of a “quantum (Hilbert) connection.” We formulate the theory in terms of the coherent state path integral. A particular interest is focused on the topological quantization in connection with the representation of compact Lie group and its application to the many-particle problem. We also examine the geometric characteristics of the canonical phase by using a spin model.
From our recently developed representation of the path integral in terms of coherent states of the dynamical group SU(1,1), we obtain the semiclassical limit as a large-NNN limit. Specifically, the parameter k=12(l+N2)k=\frac{1}{2}(l+\frac{N}{2})k=21​(l+2N​) which labels the relevant SU(1,1) representations plays the role of ${\ensuremath{\hbar}}^{\ensuremath{-}1}$. A phase-integral quantization rule is given and used to calculate the energy eigenvalues for various isotropic anharmonic oscillators. We obtain results comparable to the first-order JWKB calculations even at N=1N=1N=1.