Abstract:
An experimentally feasible realization of testing quantum-vacuum geometric phases of photons by using a gyrotropic-medium optical fibre via Casimir's effect is proposed.

Abstract:
We study the quantum-vacuum geometric phases resulting from the vacuum fluctuation of photon fields in Tomita-Chiao-Wu noncoplanar curved fibre system, and suggest a scheme to test the potential existence of such vacuum effect. Since the signs of the quantum-vacuum geometric phases of left- and right- handed (LRH) circularly polarized light are just opposite, the total geometric phases at vacuum level is inescapably absent in the fibre experiments performed previously by other authors. By using the present approach where the fibre made of gyroelectric media is employed, the quantum-vacuum geometric phases of LRH light cannot be exactly cancelled, and may therefore be achieved test experimentally.

Abstract:
The connection between the quantum-vacuum geometric phases (which originates from the vacuum zero-point electromagnetic fluctuation) and the non-normal product procedure is considered in the present Letter. In order to investigate this physically interesting geometric phases at quantum-vacuum level, we suggest an experimentally feasible scheme to test it by means of a noncoplanarly curved fiber made of gyrotropic media. A remarkable feature of the present experimental realization is that one can easily extract the nonvanishing and nontrivial quantum-vacuum geometric phases of left- and/or right- handed circularly polarized light from the vanishing and trivial total quantum-vacuum geometric phases.

Abstract:
The propagation of monomode photons inside a coiled optical fibre was regarded as a time-dependent quantum evolution process, which gives rise to a geometric phase. It is well known that the investigation of non-adiabatic geometric phases ought to be performed only in the Schr\"{o}dinger picture. So, the projections of photon spin operators onto the fixed frame of reference is discussed in this paper. In addition, we also treat the non-normal-order spin operators and consider the potential effects (e.g., quantum-vacuum geometric phases) of quantum fluctuation fields arising in a curved optical fibre. The quantum-vacuum geometric phase, which is of physical interest, can be deducted by using the operator normal product, and the doubt of validity and universality for the normal-normal procedure applied to time-dependent quantum systems is thus proposed. In the Appendix, the discussion of possible experimental realizations of quantum-vacuum geometric phases is briefly presented.

Abstract:
On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a transformation group.

Abstract:
In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also introduce a number of examples that will help the reader understand the basic issues involved. In the second part we show how to perform a universal quantum computation using only geometric effects appearing in quantum phases. It is then finally discussed how this geometric way of performing quantum gates can lead to a stable, large scale, intrinsically fault-tolerant quantum computer.

Abstract:
A quantum object can accumulate a geometric phase when it is driven along a trajectory in a parameterized state space with non-trivial gauge structures. Inherent to quantum evolutions, a system can not only accumulate a quantum phase but may also experience dephasing, or quantum diffusion. Here we show that the diffusion of quantum trajectories can also be of geometric nature as characterized by the imaginary part of the geometric phase. Such an imaginary geometric phase results from the interference of geometric phase dependent fluctuations around the quantum trajectory. As a specific example, we study the quantum trajectories of the optically excited electron-hole pairs, driven by an elliptically polarized terahertz field, in a material with non-zero Berry curvature near the energy band extremes. While the real part of the geometric phase leads to the Faraday rotation of the linearly polarized light that excites the electron-hole pair, the imaginary part manifests itself as the polarization ellipticity of the terahertz sidebands. This discovery of geometric quantum diffusion extends the concept of geometric phases.

Abstract:
A sequence of completely positive maps can be decomposed into quantum trajectories. The geometric phase or holonomy of such a trajectory is delineated. For nonpure initial states, it is shown that well-defined holonomies can be assigned by using Uhlmann's concept of parallel transport along the individual trajectories. We put forward an experimental realization of the geometric phase for a quantum trajectory in interferometry. We argue that the average over the phase factors for all quantum trajectories that build up a given open system evolution, fails to reflect the geometry of the open system evolution itself.

Abstract:
We introduce a simple experiment involving a double-slit interferometer by which one can learn basic concepts of quantum interference such as which-path marking, quantum erasers, and geometric phases. Each of them exhibits seemingly mysterious phenomena in quantum physics. In our experiment, we use the double-slit interference of visible light with the polarization as an internal state to demonstrate the disappearance of fringes by which-path marking, recovery of interference using quantum erasers, and the rapid shifting of the fringe pattern induced by the geometric phase. We also present a simple theoretical analysis of an interferometer with an internal state.

Abstract:
Calculation aspects of holonomic quantum computer (HQC) are considered. Wilczek--Zee potential defining the set of quantum calculations for HQC is explicitly evaluated. Principal possibility of realization of the logical gates for this case is discussed.