A universal formalism to describe geometric phases in non-Hermitian quantum mechanics was proposed recently by Cui Xiaodong and Zheng Yujun, researchers in the School Physics at Shandong University. The universal description of geometric phases and their characteristic natures in a non-Hermitian setting have not been in progress for more than twenty years. Their research on this question was published in Physical Review A (Phys. Rev. A 86, 064104 (2012)), a journal of the American Physics Society.

There are evident differences between non-Hermitian quantum mechanics and (standard) quantum mechanics. Non-Hermitian quantum mechanics does not obey the first axiom of quantum mechanics in general, i.e., to each non-Hermitian quantum system a complex Hilbert space may not be associated. In other words, its state space is not isometric and isomorphic to its dual space, on which the inner product can not be defined similar with in complex Hilbert space whose dual space is itself.

Fruthermore, a non-Hermitian quantum system is governed by a non-Hermitian Schrödinger equation, which results in a non-unitary evolution of its state. In their method, the researchers induce a linear functional naturally by employing the adjoint non-Hermitian Schrödinger equation.

Their formalism realizes an analog to inner product defined on a complex Hilbert space. Moreover, by generalizing the concept of interference, the generalized Bargmann invariants are therefore obtained, which is the key to describing geometric phases in non-Hermitian quantum mechanics.

Geometric phases in quantum mechanics originate from the geometric structures of parameter manifolds or complex Hilbert spaces. Geometric phases aroused widespread attention because of their topological invariance which can resist the disturbance caused by motions. Among several solutions for quantum computation, geometric quantum computation is designed according to these characteristics.

The investigation of geometric phases in non-Hermtian quantum mechanics not only allows us to deepen our understanding of the geometric structures of quantum state space and its dual space, but also has potential applications in the areas of geometric quantum computations.This work was supported by the National Science Foundation of China.

Written by: Cui Xiaodong

Edited by: Ben Hammer, Jing Zizhao

Source: School of Physics,www.view.sdu.edu.cn