New Anatomy of Quantum Holonomy

The interest in Berry’s geometric phase is now two-fold. Theoretical interest to this quintessentially quantum phenomenon is well documented [1]. The central importance is in the appearance of the gauge structure in parametric space [2–4]. Immediate utility of geometric phase is recently highlighted by the suggestion of holonomic and adiabatic quantum computings [5–9]. Consider the change of a stationary quantum state by an adiabatic variation of a system parameter. When the variation of the parameter is cyclic, namely it comes back to the original value, the state does not necessarily returns exactly to the original stationary state. This phenomenon, known as quantum holonomy, comes in three flavors, Berry phase, WilczekZee holonomy, and spiral eigenvalue holonomy. In case of Berry phase, a stationary state comes back to itself with extra phase added after an adiabatic cyclic parameter variation, and in case of Wilczek-Zee holonomy, a state belonging to a set of degenerate eigenstates turns into a mixing of those states sharing the same energy. In case of spiral holonomy, on the other hand, an eigenstate evolves into another eigenstate with different energy after an adiabatic cyclic parameter variation [10–13]. Compared to the well-known first two, which has been thoroughly studied, the newly-found spiral holonomy has been something of an odd man out, since it is yet to find its proper place in the general formulation of quantum holonomies in terms of gauge connections, and this might be the reason for the lack of due recognition and appreciation of the existence of the spiral holonomy. This is deplorable for practical reason in understanding the mechanism of spiral holonomy, as well as it is theoretically unsatisfactory, since spiral holonomy has obvious advantages in both control robustness and variety of achievable states with adiabatic quantumstate control, which enhance its usefulness in quantum computing [13]. In this paper, we intend to solve the problem of making sense of spiral holonomy, and prove that it indeed belongs to the category of quantum holonomy. We show that the non-Abelian Mead-Berry gauge connection is again the key in developing a unified formalism which treats all the quantum holonomy in a single fold. The resulting formula also sheds new light on the gauge invariance of quantum holonomy and its implications. We start by considering a quantum system described by a Hamiltonian which depends on sets of parameters, which are collectively referred to as α. Let us assume that the parameter is temporally varied α = α(t) = αt. Quantum evolution is described by the Schrödinger equation