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Journal of Physics A Mathematical and General

Geometrical phase in the cyclic evolution of non-Hermitian systems

G. Dattoli, R Mignani, A. Torre

Published Dec 21, 1990• 100 citations
Publisher

Abstract

The authors derive, by a biorthonormal state approach, the analogy of Berry's phase factor for open, non-conservative systems, for both adiabatic and non-adiabatic evolution. In the latter case, a (non-unitary) evolution operator method is exploited. An application is given to the optical supermode propagation in the free-electron laser.

DOI
10.1088/0305-4470/23/24/020

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The authors derive, by a biorthonormal state approach, the analogy of Berry's phase factor for open, non-conservative systems, for both adiabatic and non-adiabatic evolution. In the latter case, a (non-unitary) evolution operator method is exploited. An application is given to the optical supermode propagation in the free-electron laser.
In a previous paper, the measurement of an atomic Berry phase associated with two crossing levels has been suggested as a possible test for an atomic interferometry method. As one of the two levels involved is radiative, an analysis of geometrical phase factor arising in the cyclic evolution of a non-Hermitian Hamiltonian is undertaken here. It is shown that the use of the dual basis allows a simple generalization of Berry's results and that, in addition to the two possible values encountered in the Hermitian case (0, π), the geometrical phase can take on as intermediate value π/2 or even a complex value, whereas the corresponding quantum states may be permuted. The validity of the adiabatic approximation is discussed in view of the measurability of this new effect.
In a quantum system initially in the nnn-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized inverted harmonic oscillator. Although a pseudo-inner product can be introduced to make resonance states formally normalizable, this procedure relies on a non-unitary metric operator and defines a modified Hilbert space; it does not restore normalizability nor self-adjointness in the standard L2(R)L^2(R)L2(R) framework.
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:
In a quantum system initially in the n-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian with imaginary eigenvalues. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized harmonic oscillator.
The well-known geometric phase present in the quantum adiabatic evolution discovered by Berry many years ago has its analogue, the Hannay phase, in the classical domain.We calculate the Berry phase with examples for quantum hermitian and non-hermitian PTPTPT-symmetric Hamiltonians and compare with the Hannay phase in their classical equivalents.We use the analogy to propose resonant electric circuits which reproduce the theoretical solutions in simulated laboratory experiments.