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Europhysics Letters (EPL)

Geometrical Phase Factor for a Non-Hermitian Hamiltonian

Christian Miniatura, Clément Sire, J. Baudon, Jean Bellissard

Published Oct 1, 1990• 33 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/13/3/002

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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.
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.
Abstract A generalization of the adiabatic theorem for quantum systems governed by pseudo-Hermitian Hamiltonians and details of its demonstration are given. Introducing a modified time-dependent metric giving a precise description of the quantum unitary evolution where the obtained effective Hamiltonian is observable because it is mean value is real. We show that an eigenstate of a pseudo-Hermitian Hamiltonian slowly transported will acquire a real generalized geometrical phase factor which contains two contributions: the first one corresponds to the conventional Berry’s phase as expected and a new geometrical term that we call the metric geometrical phase. We will apply our results in the cases of the famous time dependent brachistochrone problem and a non-Hermitian displaced harmonic oscillator.
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.
Concepts from non-Hermitian quantum mechanics have proven useful in understanding and manipulating a variety of classical systems, such as those encountered in optics, classical mechanics, and metamaterial design. Recently, the non-Hermitian analog of the Berry phase for adiabatic processes was experimentally measured. In non-Hermitian systems, the Berry phase can have an imaginary part, which contributes to the amplification or decay of the total wave intensity. When the imaginary part of the Berry curvature is zero, this geometric amplification factor is determined solely by the initial and final points of the adiabatic path in parameter space, and it does not depend on how these points are connected by the path. We list classes of non-Hermitian Hamiltonians where this path independence is guaranteed by suitable symmetries, and we find that, for some of these classes, the amplification factor can be written only in terms of the Petermann factors of the initial and final points. Our result can, in turn, be used to experimentally obtain the Petermann factor by observing how the norm of the wave function changes under adiabatic processes. We validate our theory using a couple of concrete examples of physical relevance.