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Journal of Physics A Mathematical and General

Spin-1/2 system coupled to a single mode: a resolvent expansion of the time-dependent solution

Karl Zeile, E. Sigmund, P. Schwendimann

Published Feb 21, 1989• 1 citation
Publisher

Abstract

In order to describe the dynamics of a spin-oscillator Hamiltonian analytically, a second-order differential equation for the transition probability amplitude is established by means of a transcription into the Bargmann space of entire functions. The solution of the differential equation is given by a resolvent expansion whose zeroth-order member is in general exact up to fourth order in time; for various degenerate parameter values, however, it is exact for all times. The Laplace transform of any higher-order member is given explicitly.

DOI
10.1088/0305-4470/22/4/008

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In order to describe the dynamics of a spin-oscillator Hamiltonian analytically, a second-order differential equation for the transition probability amplitude is established by means of a transcription into the Bargmann space of entire functions. The solution of the differential equation is given by a resolvent expansion whose zeroth-order member is in general exact up to fourth order in time; for various degenerate parameter values, however, it is exact for all times. The Laplace transform of any higher-order member is given explicitly.
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We introduce a novel method to find exact density operators for a spin-1/2 particle in time-dependent magnetic fields by using the one-mode bosonic representation of su(2)su(2)su(2) and the connection with a time-dependent oscillator. As illustrative examples, we apply the method to find the density operators for constant and/or oscillating magnetic fields, which turn out to be time-dependent in general.
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