Some Reflections Concerning Geometrical Phases

Armin Uhlmann

Erik Sjöqvist, Vahid Azimi Mousolou, C. M. Canali

This note concerns the appearance of a dynamical phase in realizations of geometric phase shift gates. This dynamical phase is a global phase dependent upon the input state and is therefore irrelevant to the gate operation. Thus, the geometric nature of geometric phase shift gates is established.

Xiao‐Dong Yu, D. M. Tong

The geometric phase is a fundamental quantity characterizing the holonomic feature of quantum systems. It is well known that the evolution operator of a quantum system undergoing a cyclic evolution can be simply written as the product of holonomic and dynamical components for the three special cases concerning the Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase. However, for the most general case concerning the nonadiabatic non-Abelian geometric phase, how to separate the evolution operator into holonomic and dynamical components is a long-standing open problem. In this Letter, we solve this open problem. We show that the evolution operator of a quantum system can always be separated into the product of holonomy and dynamic operators. Based on it, we further derive a matrix representation of this separation formula for cyclic evolution, and give a necessary and sufficient condition for a general evolution being purely holonomic. Our finding is not only of theoretical interest itself, but also of vital importance for the application of quantum holonomy. It unifies the representations of all four types of evolution concerning the adiabatic/nonadiabatic Abelian/non-Abelian geometric phase, and provides a general approach to realizing purely holonomic evolution.

Harald Grosse, W. L. Kennedy

Jerrold Eldon Marsden

Gonzalo G. de Polavieja, Erik Sjöqvist

We show that a noncyclic phase of geometric origin has to be included in the approximate adiabatic wave function. The adiabatic noncyclic geometric phase for systems exhibiting a conical intersection as well as for an Aharonov–Bohm situation is worked out in detail. A spin-12 experiment to measure the adiabatic noncyclic geometric phase is discussed. We also analyze some misconceptions in the literature and textbooks concerning noncyclic geometric phases.