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Journal of Physics A Mathematical and General

Evolution of twofold degenerate two-level system. Geometrical effects

Mikhail Pletyukhov, E. A. Tolkachev

Published Jan 1, 1999• 2 citations
Publisher

Abstract

Geometrical effects occurring in the cyclic evolution of twofold degenerate two-level system are considered. A projector representation for the objects involved (Hamiltonian, evolution operator) is constructed. The explicit expressions for connections and curvatures are quoted. Particular cases are considered and found to be in full correspondence with previously obtained results.

DOI
10.1088/0305-4470/32/6/017

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Geometrical effects occurring in the cyclic evolution of twofold degenerate two-level system are considered. A projector representation for the objects involved (Hamiltonian, evolution operator) is constructed. The explicit expressions for connections and curvatures are quoted. Particular cases are considered and found to be in full correspondence with previously obtained results.
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Recently we showed that quantum cyclic evolution occurs, provided the initial wave function was one of the eigenfunctions of the evolution operator, whatever the concrete form of the Hamiltonian [Phys. Rev. A 50, 5317 (1994)]. This paper will show that, for some specific Hamiltonians, cyclic evolution may occur, even if the initial wave function is not one of the eigenfunctions of the evolution operator. The general conditions for the occurrence of cyclic evolution is derived. Several interesting examples of quantum evolution, also including noncyclic evolution, and some specific geometric phases are given. \textcopyright{} 1996 The American Physical Society.
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The geometric phase of mixed states with non-degenerate eigenvalues is investigated. A general formula of geometric phase for mixed state under unitary evolution is given. In particular, we also furnish an expression of Hamiltonians for equivalent evolutions, by which one can understand what kind of evolutional operator U(t) (or Hamiltonian) is related to zero instantaneous dynamic phase. Moreover, the geometric phase and related Hamiltonians in the spin-half case are provided as an explicit example.
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A simple systematic way of obtaining analytically solvable Hamiltonians for quantum two-level systems is presented. In this method, a time-dependent Hamiltonian and the resulting unitary evolution operator are connected through an arbitrary function of time, furnishing us with new analytically solvable cases. The method is surprisingly simple, direct, and transparent and is applicable to a wide class of two-level Hamiltonians with no involved constraint on the input function. A few examples illustrate how the method leads to simple solvable Hamiltonians and dynamics.