For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:
The method used by Garrison and Wright for the calculation of complex geometrical phases in quantal dissipative systems is extended to all time periodic linear systems, including classical linear systems with nonunitary time evolution. Each eigenvector
∣xi(t)〉 of the period map modified by a complex scaling exponent ${\ensuremath{\theta}}_{i}(t)$ has an associated complex geometrical phase, which admits a solid angle interpretation in two-dimensional systems. Pairs of the
∣xi(t)〉 commonly determine quadratic periodic invariants corresponding to topologically distinct surfaces of the invariant torus, and a connection between the scaling exponents and classical angle variables is established. The method is illustrated by the calculation of explicit expressions for the classical and quantal nonadiabatic phases associated with angular momentum precession. A second application to a general quadratic Hamiltonian extends existing adiabatic results to the classical and quantal nonadiabatic phases of both elliptic and hyberbolic systems. The relevance of torus ensemble averaging in a classical angle-action formulation is discussed.