Unification of the family of Garrison-Wright's phases.

Xiao-Dong Cui,Yujun Zheng

doi:10.1038/srep05813

Abstract

Inspired by Garrison and Wight's seminal work on complex-valued geometric phases, we generalize the concept of Pancharatnam's “in-phase” in interferometry and further develop a theoretical framework for unification of the abelian geometric phases for a biorthogonal quantum system modeled by a parameterized or time-dependent nonhermitian hamiltonian with a finite and nondegenerate instantaneous spectrum, that is, the family of Garrison-Wright's phases, which will no longer be confined in the adiabatic and nonadiabatic cyclic cases. Besides, we employ a typical example, Bethe-Lamb model, to illustrate how to apply our theory to obtain an explicit result for the Garrison-Wright's noncyclic geometric phase, and also to present its potential applications in quantum computation and information.

Highlights

Inspired by Garrison and Wight’s seminal work on complex-valued geometric phases, we generalize the concept of Pancharatnam’s ‘‘in-phase’’ in interferometry and further develop a theoretical framework for unification of the abelian geometric phases for a biorthogonal quantum system modeled by a parameterized or time-dependent nonhermitian hamiltonian with a finite and nondegenerate instantaneous spectrum, that is, the family of Garrison-Wright’s phases, which will no longer be confined in the adiabatic and nonadiabatic cyclic cases
It turns out that such a constraint on nonhermitian hamiltonian operators can conduce to effective generalization of concepts of geometric phases in conventional quantum mechanics
In this paper, we have extended the concept of Pancharatnam’s ‘‘in-phase’’ in interferometry based on generalized interference formula for biorthogonal quantum systems

Summary

D One can check that the covariant derivative can be defined as ds

D jyðsÞi~ d {AGPðsÞ jyðsÞi: ð13Þ ds ds Likewise, the duals of Eqs. (6)–(13) can be obtained by (exdDc~shya~nðgsÞindDgs y w)ith y~. Let the two non-biorthogonal states jy1æ, jy2æ be connected by any geodesic G1,2 satisfying Eq (16), the generalized. Pancharatnam connection, and indicates that the global generalized Pancharatnam’s phase difference between the initial and final states is gauge-dependent and can not be considered as the geometric phase or Garrison-Wright’s phase in a biorthogonal quantum system. A universal formula for geometric phase between the initial jy1æ and the final state jyNæ is given by cgeo~{. %. It should be noted that the proposition can be applied in calculating the nonadiabatic Garrison-Wright’s phase and in evaluating quantum speed limit (QSL)[35] from the viewpoint of geometry. The evolving state jy(t)æ of a biorthogonal quantum system governed by Schrodinger-like equation starts from an initial state jy(0)æ and ends in the non-biorthogonal final state jy(t)æ, the geometric phase cnc(0, t) between jy(0)æ and jy(t)æ is given by Eq (21)

E N ÞDt E

Reγgeo

Conclusion