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Chinese Physics Letters

Exact Solution of the Bloch Equation with Time-Dependent Coefficients and Aharonov-Anandan Phase

Jing‐Bo Xu, Lan Wang, Xiao-Chun Gao

Published Mar 1, 1993• 0 citations
Publisher

Abstract

In this letter, the matrix invariant of the Bloch equation with time-dependent coefficients is found, the time evolution of the system is discussed and the exact solution of the system is obtained. We then proceed to study the cyclic evolution of the system, obtaining the Aharonov-Anandan phase. Finally, it is shown that the study of noncyclic evolution of the system can be reduced to the cyclic evolution.

DOI
10.1088/0256-307x/10/3/001

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In this letter, the matrix invariant of the Bloch equation with time-dependent coefficients is found, the time evolution of the system is discussed and the exact solution of the system is obtained. We then proceed to study the cyclic evolution of the system, obtaining the Aharonov-Anandan phase. Finally, it is shown that the study of noncyclic evolution of the system can be reduced to the cyclic evolution.
It is found that in a quantum system with cyclic Hamiltonian the time-dependent Schrodinger equation admits the Bloch type solutions which have the form of Bloch function with respect to the time. These solutions constitute an orthonormal basis of the solution space and lead to a new type of quantum phases. The new phases, called Bloch phases by the authors, are the special Aharonov-Anandan phases, and reduce to the Berry phases under the adiabatic approximation if the degeneracy of instantaneous eigenenergy is time-independent. The above conclusions are proved and calculation for the evolution of spin magnetic moments in a magnetic field precessing with constant angular velocity is exemplified.