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Chinese Physics Letters

Parametrization of Spin Rotation and the Two-Level Problem

Jin-yuan Zhou, Li-yun Li, Chao-hui Ye

Published Mar 1, 1996• 1 citation
Publisher

Abstract

The evolution behavior of a spin system subject to a time-varying magnetic field and the relevant two-level problem essentially exhibit like a rotation in a three-dimensional space. It is shown that the parametrization of this rotation is shifted into solving the time-dependent Schrödinger equation of a spin-1/2 system and the rotation parameters can thus be obtained by a straightforward method.

DOI
10.1088/0256-307x/13/3/001

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The evolution behavior of a spin system subject to a time-varying magnetic field and the relevant two-level problem essentially exhibit like a rotation in a three-dimensional space. It is shown that the parametrization of this rotation is shifted into solving the time-dependent Schrödinger equation of a spin-1/2 system and the rotation parameters can thus be obtained by a straightforward method.
All the six kinds of rotation approach with the same form to the evolution problem of arbitrarily time-dependent two-level system are investigated in this paper. A time-dependent two-level system can be viewed as a spin-1/2 system in a time-varying magnetic field. It is shown that for each kind of rotation approach, we can always find a rotating frame in which the direction of the effective magnetic field is fixed. This property reduces the problem of finding the time-evolution operator to the solution of a second-order differential equation. Applications are made to the J C model in quantum optics and the L and au–Zener model in resonance physics.
For tutorial purposes, the simple problem of a classical magnetic moment interacting with various magnetic field configuration is reformulated in terms of coordinate-free, time-dependent rotation operators. In this approach, the corresponding equation of motion assumes the form of an operator equation, identical with the Schrödinger equation of a spin-1 interacting with the same field configurations. It is solved utilizing deliberately the concepts of a Hermitian Hamiltonian and a unitary time evolution operator to familiarize students early in their curriculum with these important ingredients of the quantum-mechanical formalism. It turns out that besides furnishing useful exercises for the manipulation of three-dimensional rotation operators, being an end in itself, the approach also provides simple but nontrivial examples for a number of theorems for which the students will have frequent use in their quantum-mechanics course.
We consider three spin systems in the parallel magnetic fields and present some of possible exact solutions.In addition, we discuss an analogue of the Rabi problem for three-spin system.In order to do this, we reduce the problem of three coupled spins, an 8-level system, to a problem of a 3-level system in an effective field.We explicitly construct the evolution operator of the system in two special cases: when the fields and interaction are the same in all the spins but with a arbitrary time-dependence of the fields, and when the spins are subjected to a circular magnetic field (the Rabi problem).Some applications of these results are discussed.
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The time-dependent Schr\"odinger equation is solved formally for an atomic or molecular system which is subjected simultaneously to a rotating magnetic field of constant amplitude and angular velocity and to a constant magnetic field along the axis of rotation. The method yields the transition probabilities in terms of the solutions to an eigenvalue problem. This eigenvalue problem is solved both for (a) a normal Zeeman effect and for (b) the case where a transition from a given level is isolated in frequency from other transitions from the same level. Case (a) is exactly soluble and yields a solution which is shown to be the same as that of Bloch and Rabi, but is in a form which is more convenient for integration over the velocity distribution. Case (b) must be solved by an approximate method which results in a prediction of multiple quantum transitions as observed by Kusch.